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....

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.