What challenges faced Europe in the time from the 1960s to the 1990s?By the 1960s, much of Europe had recovered from the effects of World War II. Yet, new issues would confront the continent.
This is a huge question, well beyond the scope of a good answer in this space. Different issues confronted different countries/regions of Europe at different times in the 40 year period you are asking about.
I would argue that the two major issues during this time (depending, again, on the country/region) were issues of economic growth and issues of Cold War politics.
Economic growth is an issue for all countries at all times. It was particularly important in Europe during this time since Europe was still trying to recover from WWII and break the US dominance of the economic world. In Eastern Europe, the problem was more related to communism as the communist governments tried (and failed) to find a way to give their citizens a decent standard of living.
Cold War issues surfaced in all areas. There were, for example, tensions between Western European countries and the US over things like nuclear weapons. There were worries in the Eastern Bloc about how best to protect and expand their system.
However, there were of course many more challenges that faced Europe at one or more times and places during this time.