  We study the effects of short-range interactions on a generalized
three-dimensional Weyl semimetal, where the band touching points act as the
(anti)monopoles of Abelian Berry curvature of strength $n$. We show that any
local interaction has a \emph{negative} scaling dimension $-2/n$. Consequently
all Weyl semimetals are stable against weak short-range interactions. For
sufficiently strong interactions, we demonstrate that the Weyl semimetal either
undergoes a first order transition into a band insulator or a continuous
transition into a symmetry breaking phase. A translational symmetry breaking
axion insulator and a rotational symmetry breaking semimetal are two prominent
candidates for the broken symmetry phase. At the one-loop order, the
correlation length exponent for continuous transitions is $\nu=n/2$, indicating
their non-Gaussian nature for any $n>1$. We also discuss the scaling of the
thermodynamic and transport quantities in general Weyl semimetals as well as
inside the broken symmetry phases.
