Much to Turkey's continued annoyance, transport through the Bosphorus, the Sea of Marmara, and the Dardanelles (that is, the passage from the Black Sea to the Mediterranean) is covered by the 1936 Montreux Convention, also called the "Turkish Straits Convention".

The "Straights Question" was of continual importance to Russia, and later the Soviet Union, during the 19th and 20th Centuries. After World War I, the 1923 the Treaty of Lausanne eventually officially demilitarized the straights and opened the Straits to unrestricted civilian and military traffic under the control of an International Straits commission. The 1936 Montreux Convention built on this, and was meant in part to check Italy (who at the time controlled several Greek islands near the Dardanelles). It returned military control to Turkey, allowed Turkey to refortify the straits (remember, the hardest fighting in the Ottoman had been around Galipoli, which was the British attempt to take the Dardanelles).

Montreux allows complete free passage to merchant ships during peace time, and almost free passage to military vessels during peace time. There are restrictions on tonnage, number, and duration of stay for non-Black Sea nations but Black Sea nations, including the USSR, have the right to unrestricted transit. There are also special clauses for submarines, aircraft carriers, etc (more than half of the convention is about military ships, with civil ships and all other matters being covered else where), but the principle was that Black Sea nations (which meant, effectively, Russia since no other Black Sea country beside Turkey had a significant navy) had more or less free passage during peace time, and Turkey got to legally refortify the straits, which was an improvement over Lausanne.

One of the reasons Turkey was so eager to join NATO was because of the Straits Question (another big reason was that the fear that the Soviets would more actively claim eastern Anatolian cities like Kars, which they were actually ready to do in the 1950's). Turkey stayed neutral all through WWII, but accepted Truman's Marshall Plan money in 1946 in large part because of the Turkish Straits Crisis, and later joined NATO in 1952 as a further hedge against the Soviet Union. In subsequent years, the Soviet Union would sometimes try to use the Montreux Convention to limit American passage through the straits since America, a non-Black Sea nation, had more limited rights under the Convention.

Montreux is still in effect today. Orhan Pamuk, in his book Istanbul, says that one of his clearest memories from his childhood is watching a huge Russian battleship come through the Bosphorus. A few years ago I was having breakfast by the Bosphorus and saw a massive Russian submarine (on the surface) come through the Bosphorus. In fact, Turkey has refused to sign the 1994 UN Convention on the Law of the Sea in large part to keep Montreux in place (though I'm not sure of exactly what they would have to give up if they did sign). This is beyond this subreddit's 20-year rule, so I will only mention it briefly, but Turkey is now building a "Second Bosphorus" called "Kanal Istanbul" for reasons which I quite frankly don't understand, but seems likely linked to getting around some part Montreux.

And just one additional note: the Bosphorus is really the lifeline of Istanbul (I cross it several times a week either by bridge, ferry, or the recently opened subway tunnel), and historically much of the growth of Istanbul has been along its banks. I'm unclear exactly when it happened, but at some point during the Cold War as a precaution, the Turkish military opened up a lot of the area around the Bosphorus by knocking down existing structures and claiming imminent domain. There are some areas where this didn't happen (particularly historically important areas), but in generally most of the coastline of the Bosphorus is now park land (which of course is open enough for military vehicles in war time).

Sort of an add-on question. How often or easy was it for the Soviets to use canals and rivers, or possibly portaging, between the Baltic, White, and Black seas for their ships. In other words, how necessary was the Bosphorus for transfer of their ships between theaters, and how viable were the alternatives?