Before answering, it's worth noting that containment is a fair description of the general US strategy in the Cold War, but it is difficult to define what exactly it means. Every Cold War president defined it differently and implemented a policy aimed at achieving their end. Truman saw the original version of it and expanded its scope from Europe into Asia. Eisenhower used the "New Look" which relied on brinkmanship and the nuclear deterrent. Kennedy (and to some extent Johnson) employed "Flexible Response" which relied more heavily on military and civil aid. Nixon and Ford used detente and great power politics. Carter is a bit tougher to pin do but tried to use a mixture of traditional measures and international consensus. Reagan pursued "Peace through Strength," and sought to raise the economic cost of the Cold War. Even George Kennan, the author of the doctrine, is notoriously inconsistent in how he defines it. For the purpose of this answer, I will use the a definition of limiting Soviet influence and activity, rather than some of the more aggressive ones that employed ideas of rollback.

Based on that criteria, containment was unambiguously successful in Europe. Actions like the creation of NATO, the Marshall Plan, and the Berlin Airlift demonstrated significant western resolve very early in the Cold War and showed the significant risk involved in communist agitation there. As a result, there was little effective Soviet-backed agitation in western Europe after the 1940s. The exception was Berlin, which was in a state of more or less perpetual crisis until the construction of the Berlin Wall. However, once the Wall went up both sides used it rhetorically but also recognized that it solved the immediate reasons for overt hostile action.

Containment was also arguably successful in Northeast Asia. Korea remained divided after the Korean War more or less along the lines of the identified zones of control and there was never a real communist threat to Japan. Containment was a disaster in Southeast Asia as evidenced by Vietnam. This was the most notable and costly failure of the policy. Additionally, containment failed miserably in Africa, creating conflict in Angola, Rhodesia, Ethiopia, and Somalia. Finally, the policy was largely unsuccessful in Latin and South America as the region remained one of the more active battlegrounds of the Cold War from the mid-70s through the end of the conflict.

However, overall I think it is fair to call the policy a success. Europe was the most important strategic location for the US, and the only location that the US would have without a doubt resorted to nuclear weapons to save. The resolve that NATO showed there caused the conflicts of the Cold War to migrate to other regions of the world, where failure was embarrassing but not existential to the US or USSR. However, this proved devastating for many of those countries and cost hundreds of thousands to millions of lives as a result.

If you are looking for good reads on the subject, I'd recommend Odd Arne Westad's Global Cold War and John Lewis Gaddis' The Cold War. Both are well researched and well-written accounts that show the full scope of the conflict.