  We present new results on equisingularity and equinormalizability of families
with isolated non-normal singularities (INNS) of arbitrary dimension. We define
a $\delta$-invariant and a $\mu$-invariant for an INNS and prove necessary and
sufficient numerical conditions for equinormalizability and weak
equinormalizability using $\delta$ and $\mu$. Moreover, we determine the number
of connected components of the Milnor fibre of an arbitrary INNS. For families
of generically reduced curves, we investigate the topological behavior of the
Milnor fibre and characterize topological triviality of such families. Finally
we state some open problems and conjectures. In addition we give a survey of
classical results about equisingularity and equinormalizability so that the
article may be useful as a reference source.
