  A comprehensive theory of the quantum phase transition in clean, itinerant
Heisenberg ferromagnets is presented. It is shown that the standard mean-field
description of the transition is invalid in spatial dimensions $d\leq 3$ due to
the existence of soft particle-hole excitations that couple to the order
parameter fluctuations and lead to an upper critical dimension $d_c^+ = 3$. A
generalized mean-field theory that takes these additional modes into account
predicts a fluctuation-induced first-order transition. In a certain parameter
regime, this first-order transition in turn is unstable with respect to a
fluctuation-induced second-order transition. The quantum ferromagnetic
transition may thus be either of first or of second-order, in agreement with
experimental observations. A detailed discussion is given of the stability of
the first-order transition, and of the critical behavior at the
fluctuation-induced second-order transition. In $d=3$, the latter is mean
field-like with logarithmic corrections to scaling, and in $d<3$ it can be
controlled by means of a $3-\epsilon$ expansion.
