There are at least 16 cases of Marvel/DC crossover, beginning with “Superman vs. The Amazing Spider-Man” in 1976. Other teamups include Spider-Man & Batman, X-Men & Teen Titans, Daredevil & Batman, and JLA & Avengers. You can see a list here.
In addition, the two companies have often spoofed and 'borrowed' from each other. Marvel introduced the Squadron Sinister as a thiny veiled, villanous version of Justice League. Squadron Sinister later got a good, parallel universe version called Squadron Supreme. The first version of the Shi'ar Imperial Guard had quite a lot in common with DC's Legion of Super-Heroes (probably because the X-Men artist at the time, Dave Cockrum, had drawn LSH for years). DC's General Glory was, in return, originally a spoof on Captain America. There are probably many more such examples.