  There are theoretical and phenomenological motivations that there may exist
additional heavy Z' bosons with family non-universal couplings. Flavor mixing
in the quark and lepton sectors will then lead to flavor changing couplings of
the heavy Z', and also of the ordinary Z when Z-Z' mixing is included. The
general formalism of such effects is described, and applications are made to a
variety of flavor changing and CP-violating tree and loop processes. Results
are described for three specific cases motivated by a specific heterotic string
model and by phenomenological considerations, including cases in which all
three families have different couplings, and those in which the first two
families, but not the third, have the same couplings. Even within a specific
theory the results are model dependent because of unknown quark and lepton
mixing matrices. However, assuming that typical mixings are comparable to the
CKM matrix, processes such as coherent mu-e conversion in a muonic atom,
K^0-\Bar{K}^0 and B-\Bar{B} mixing, epsilon, and epsilon'/epsilon lead to
significant constraints on Z' bosons in the theoretically and
phenomenologically motivated range M_{Z'} \sim 1 TeV.
