  The Bochner tensor is the K\"ahler analogue of the conformal Weyl tensor. In
this article, we derive local (i.e., in a neighbourhood of almost every point)
normal forms for a (pseudo-)K\"ahler manifold with vanishing Bochner tensor.
The description is pined down to a new class of symmetric spaces which we
describe in terms of their curvature operators. We also give a local
description of weakly Bochner-flat metrics defined by the property that the
Bochner tensor has vanishing divergence. Our results are based on the local
normal forms for c-projectively equivalent metrics. As a by-product, we also
describe all K\"ahler-Einstein metrics admitting a c-projectively equivalent
one.
