  This paper has two main results. Firstly, we complete the parametrisation of
all p-blocks of finite quasi-simple groups by finding the so-called
quasi-isolated blocks of exceptional groups of Lie type for bad primes. This
relies on the explicit decomposition of Lusztig induction from suitable Levi
subgroups. Our second major result is the proof of one direction of Brauer's
long-standing height zero conjecture on blocks of finite groups, using the
reduction by Berger and Kn\"orr to the quasi-simple situation. We also use our
result on blocks to verify a conjecture of Malle and Navarro on nilpotent
blocks for all quasi-simple groups.
  In the supplement we treat a configuration for the $5$-blocks of the finite
simple groups $E_8(q)$ that was inadvertently omitted in the first version and
clarify and correct the meaning of some entries in Tables 3 and 4. The main
theorems of the original version are not affected.
