The Soviet Union had anti-tank weapons that could knock out an Abrams.

In particular, Houthi rebels in Yemen have knocked out multiple Saudi Abrams tanks using the Iranian Tosan wire-guided missile. This is an Iranian license-built version of the Soviet-era 9M113 Konkurs missile (NATO reporting name AT-5 Spandrel). This wasn't made specifically to counter the Abrams - it went into service in 1976, years before the Abrams. The frontal armour of an Abrams should be able to stop it, but it has proven capable of knocking out the Abrams when hitting the side, rear, or top armour. (Modern Russian tanks, including the Soviet-era T-80, have frontal armour that is probably less effective than that of the Abrams, but have resisted missiles such as the TOW (which has a warhead about double the weight that of the Konkurs).

This general principle of "frontal armour stops it, but not the sides/rear/top" applies to other weapons, including Soviet-era tank guns and other anti-tank missiles. Notably, during its time in service, the Abrams has received armour upgrades designed to improve its resistance to RPGs more modern than the RPG-7, a clear sign that the original armour protection was felt to be inadequate.

Soviet/Russian tanks such as the T-80 and T-90 are, in principle, capable of taking on the Abrams, with frontal hull and turret armour that should stop APFSDS shot from the Abrams, at least beyond short range. The T-80/T-90 and the Abrams both have frontal armour that resists each other's guns, but have guns capable of knocking the opposing tank out through non-frontal hits. The vulnerability of the Abrams to non-frontal hits by modern tank guns is demonstrated by the use of the Abrams to destroy fellow Abrams tanks that would otherwise have had to be abandoned in condition where they could be recoverable by the enemy. The best-armoured versions of the Abrams should have somewhat better armour than the T-80 and T-90, but an engagement would probably be decided by tactics, operator skill, and numbers rather than this slight superiority. While there have been almost purely tank-vs-tank battles, this is rarely the best way to use armour, and a combined-arms force with an equal number of tanks can easily have a huge advantage over a tank-only force. Thus, a tank-vs-tank comparison isn't so useful.

Other weapons that can KO an Abrams include mines (and IEDs), heavy artillery (say, 150mm and up), and various missiles. For example, the 1970s Kh-23 Grom air-to-ground missile (in NATO-speak, AS-7 Kerry) with its over-100kg warhead, would be devastating to an Abrams.

Given the non-invulnerability of the Abrams, why did the Abrams perform so well in the 1991 Gulf War? The opposing Iraqi tanks lacked modern night-vision systems, and the major tank battles took place mostly at long range, with visibility reduced by night and/or smoke. Also, the opposing tanks were older-generation Soviet tanks. (The M60A1, which was still used by the US Marines, also performed very well in the fighting in Kuwait, despite facing heavier defences.)

The Abrams has performed less one-sidedly-well in the Middle East this century, with the Iraqi Army losing dozens of their US-supplied Abrams tanks to ISIL (with perhaps 9 of them later being captured by Iranian-backed militias who then proceeded to use them against ISIL). Saudi, Iraqi, and US experience in the 2003 invasion of Iraq and the post-invasion insurgency all show that the Abrams is vulnerable to tactically-adept infantry with modern anti-tank weapons. Tactically-inept use of the Abrams makes it much more vulnerable to such opposition. As already said above, tactics and skill are very important when using tanks.

There are two major and incredibly popular misconceptions here. One is of a technical nature and one of military science in general. The military science or tactical misconception first, I think. Technological objects like tanks do not exist in a vacuum, but must be considered in their intended role and use. Military technological objects - like tanks - especially so. Tanks do not exist to be plopped down in a 10km by 10km box in which they will duke it out, and whoever has the best numbers (gun, armor, optics, etc) wins. Tanks exist to accomplish specific military tasks. These tasks are dictated by what we call doctrine. Doctrine is, in the Russian approach to war, a highly systematized set of views and theories on the conduct of war, intended to produce a coherent “way of war” throughout the entire armed forces. In Soviet doctrine of the 1980s, tanks were considered to:

[C]onstitute the main striking power of the ground forces...capable of exploiting the effects of nuclear and fire strikes to the fullest, and of achieving the final objectives of an engagement or operation in short periods of time. The fighting capabilities of tank formations...make it possible for them to conduct aggressive offensive operations by day and by night when considerably separated from other forces, to defeat the enemy in meeting engagements, and to negotiate extensive zones of radioactive contamination and water obstacles from the line of march...Tanks are an important means for increasing the dynamic and mobile nature of modern combat.*

In short, at all levels of combat, the primary role of the tank in Soviet doctrine is maneuver. Aggressive maneuver to exploit penetrations created by the other branches of the army, all in order to achieve some operational goal or effect, like cutting off a retreat or seizing a politically important city. These tanks did not maneuver in a vacuum, either. They were supported by, and acting in support of, an entire combined arms force. This means mechanized infantry in armored carriers; it means artillery; it means combat aviation and air assault forces. In particular, the mechanized infantry and the tanks themselves cooperated very tightly. The tanks support the infantry with their firepower, and the infantry supports the tanks with their own firepower, especially large numbers of powerful anti-tank guided missiles - as the other commenter here described in more depth. I should note that this is also the role that most other armies assign to their tanks - certainly it is and was the case for the US Army at the time. Although the Americans also envisioned a more defensive kind of maneuver for their tank formations: counter-maneuvering to cut off Soviet armored spearheads and stop them from achieving the goals of their own maneuvers.

Why does this doctrine stuff matter? Because doctrine and technological design should go hand in hand. When we evaluate how “good” a piece of technology like a tank was, we should evaluate how well it aligned with the doctrine it was supposed to support. In the case of the T-72, the answer is, “pretty well”. The detailed answer ties into the second popular misconception I mentioned: the systemic under-estimation of Soviet armor quality in American popular writing. This under-estimation, I should point out, was not at all shared by the US Army tank community of the time. ARMOR magazine, the official publication and discussion board for the US Army armor branch, was full of worried speculation and design analysis of the latest Soviet tanks all through the late 1970s and 1980s. Brace yourself, because there's no good way to answer this part without a lot of abbreviations and technical stuff.

In actual technical fact, many T-72 models which were contemporary to the original Abrams (so-called “Improved” T-72A or T-72A obr. 1983 for instance, compare with M1 Abrams introduced ca. 1980) were qualitatively well-matched. The original M1 Abrams is, despite reputation, not fantastically well armored, although it was well armed for the time. The armor reference threats and defeat velocities are not publicly known as far as I’m aware, but it is known that as late as 1973 the TACOM design committee considered an unspecified Soviet 115mm APFSDS round and a 5 inch HEAT warhead as the protection standard for the armor. Whether this changed after their meeting with the British that same year to discuss the new “Chobham” special armor, I do not know. The 115mm round is from the prior generation of Soviet tanks (T-62/early T-64), whereas by 1967 more advanced T-64A models were already equipped with early 125mm guns. Technical specifics aside, the plain M1 Abrams model was not - as far as I know - designed to be invulnerable to its contemporary Soviet opponents. In particular, it would likely have fared poorly against the 125mm APFSDS rounds until the adoption of the Heavy Armor and Heavy Common depleted uranium armor packages in 1988-89. Meanwhile, the Soviets were significantly further along in their development of advanced armor designs than the Americans believed at the time. The performance and design of the T-72B (1986) turret armor were almost totally unknown until the early 2000s, for instance. Declassified Swedish tests of an armor package roughly comparable to the M1A1 (1985, pre-depleted uranium armor components) are linked here for more detailed comparison.

In terms of armament, this is where the Abrams starts to outdo the Soviet tanks. The 105mm gun on the original M1 was quite potent against all T-64 models and the original model T-72, owing to the advanced armor penetrator designs the US was able to field. Best open source data suggests that the 105mm could kill T-72 and probably T-72A (pre-1983) from the front at all reasonable combat distances. Monobloc tungsten and uranium APFSDS rounds (M735, M774, M833) were fielded in significant numbers as early as 1978, while the Soviets struggled right until the end of the Cold War to field comparable rounds in large quantity. The introduction of the 120mm gun on the M1A1 (1985-86) only extended this disparity. However, given the modest armoring of the original M1 and early M1A1, it is sort of a wash in terms of each tank’s ability to kill the other at combat distances. The Soviet introduction of reactive armor complexes in the late 1980s shifted this balance back out of the Abrams’ favor, however. Both the Abrams and the T-72 had fairly good optics and fire control for the time, so detection of targets and accurate engagement would be roughly comparable. The T-72’s night optics were rather lackluster, though, and the necessity of an infrared illuminator rendered it vulnerable to detection.

In short: The mainstay Soviet tanks of the late Cold Were were competently designed, produced with generally good workmanship, and could have been expected to carry out their missions against contemporary NATO tanks on a mostly even footing, all else being equal. (Which is never true in war, but you asked about tanks, so you get a tank answer). They were armored well enough to survive many of the anti-tank weapons which a NATO infantry or armored battalion could throw at them. (From the front, at combat distances, of course.) This means that on average, units could be expected to take acceptable casualties, rather than becoming combat ineffective before meeting their goals. It was fast enough over most terrain to achieve good tactical and operational mobility. In general it appears to have been reliable enough to carry out long (>500km) maneuvers without breakdown, even under conditions. The one major problem was Soviet industry’s inability to produce large quantities of advanced APFSDS rounds, which by the late 1980s did lead to a growing gap between the killing power of Soviet guns and the increasingly strong protection of NATO tanks. But - remember, in the combined arms world, you have plenty of other tools at your disposal besides the tank's own gun.

*From V. Reznichenko, Taktika (1984). Note that this outlook applies to both tactical combat - combat at the scale of battalions and divisions, usually consisting of single battles over specific objectives - and the operational scale, which is the scale of multiple divisions or corps, consisting of many coordinated tactical actions all advancing a particular goal.