From series 2, The Cosby Show made an event of their opening credits (I won't post them here), which I loved.
Series 2-4 were of a similar theme but 5-7 were VERY different. (They had to pull the original series 7 titles due to a copyright issue and but a tweaked version was used for the final series 8).
TCS spawned a spin-off series, A Different World, based around daughter Denise (my favourite) going away to college. I lost interest after series 1 because Denise (Lisa Bonet) and Maggie (Marisa Tomei) left but series 2 opening titles were notable because the theme song was sung by Aretha Franklin, no less.