  We prove that Brauer's Height Zero Conjecture holds for p-blocks of finite
groups with metacyclic defect groups. If the defect group is nonabelian and
contains a cyclic maximal subgroup, we obtain the distribution into p-conjugate
and p-rational irreducible characters. Then the Alperin-McKay Conjecture
follows provided p=3. Along the way we verify a few other conjectures. Finally
we consider the extraspecial defect group of order p^3 and exponent p^2 for an
odd prime more closely. Here for blocks with inertial index 2 we prove the
Galois-Alperin-McKay Conjecture by computing k_0(B). Then for p\le 11 also
Alperin's Weight Conjecture follows. This improves some results of [Gao, 2012],
[Holloway-Koshitani-Kunugi, 2010] and [Hendren, 2005].
