  Let $K$ be an algebraically closed field of characteristic $0$. For $m\geq
n$, we define $\tau_{m,n,k}$ to be the set of $m\times n$ matrices over $K$
with kernel dimension $\geq k$. This is a projective subvariety of
$\bbP^{mn-1}$, and is called the (generic) determinantal variety. In most cases
$\tau_{m,n,k}$ is singular with singular locus $\tau_{m,n,k+1}$. In this paper
we give explicit formulas computing the Chern-Mather class ($c_M$) and the
Chern-Schwartz-MacPherson class ($c_{SM}$) of $\tau_{m,n,k}$, as classes in the
projective space. We also obtain formulas for the conormal cycles and the
characteristic cycles of these varieties, and for their generic Euclidean
Distance degree. Further, when $K=\bbC$, we prove that the characteristic cycle
of the intersection cohomology sheaf of a determinantal variety agrees with its
conormal cycle (and hence is irreducible).
  Our formulas are based on calculations of degrees of certain Chern classes of
the universal bundles over the Grassmannian. For some small values of $m,n,k$,
we use Macaulay2 to exhibit examples of the Chern-Mather classes, the
Chern-Schwartz-MacPherson classes and the classes of characteristic cycles of
$\tau_{m,n,k}$.
  On the basis of explicit computations in low dimensions, we formulate
conjectures concerning the effectivity of the classes and the vanishing of
specific terms in the Chern-Schwartz-MacPherson classes of the largest strata
$\tau_{m,n,k}\smallsetminus \tau_{m,n,k+1}$.
  The irreducibility of the characteristic cycle of the intersection cohomology
sheaf follows from the Kashiwara-Dubson's microlocal index theorem, a study of
the `Tjurina transform' of $\tau_{m,n,k}$, and the recent computation of the
local Euler obstruction of $\tau_{m,n,k}$ .
