  It is known that, generically, Taylor series of functions holomorphic in the
unit disc turn out to be universal series outside of the unit disc and in
particular on the unit circle. Due to classical and recent results on the
boundary behaviour of Taylor series, for functions in Hardy spaces and Bergman
spaces the situation is essentially different. In this paper it is shown that
in many respects these results are sharp in the sense that universality
generically appears on maximal exceptional sets. As a main tool it is proved
that the Taylor (backward) shift on certain Bergman spaces is mixing.
