  For $X$ a connected finite simplicial complex we consider $\Delta^d(X,n)$ the
space of configurations of $n$ ordered points of $X$ such that no $d+1$ of them
are equal, and $B^d(X,n)$ the analogous space of configurations of unordered
points. These reduce to the standard configuration spaces of distinct points
when $d=1$. We describe the homotopy groups of $\Delta^d(X,n)$ (resp.
$B^d(X,n)$) in terms of the homotopy (resp. homology) groups of $X$ through a
range which is generally sharp. It is noteworthy that the fundamental group of
the configuration space $B^d(X,n)$ abelianizes as soon as we allow points to
collide (i.e. $d\geq 2$).
