  We give an introduction to the theory of $V$-filtrations of Malgrange and
Kashiwara. After discussing the basic properties of this construction (in the
case of a smooth hypersurface and, later, in the general case), we describe the
connection with the theory of $b$-functions. As an example, we treat the case
of weighted homogeneous isolated singularities. We discuss the compatibility of
$V$-filtrations with proper push-forward and duality and the connection with
nearby and vanishing cycles via the Riemann-Hilbert correspondence. We end by
describing some invariants of singularities via the $V$-filtration.
