  We develop general techniques for computing the fundamental group of the
configuration space of $n$ identical particles, possessing a generic internal
structure, moving on a manifold $M$. This group generalizes the $n$-string
braid group of $M$ which is the relevant object for structureless particles. In
particular, we compute these generalized braid groups for particles with an
internal spin degree of freedom on an arbitrary $M$. A study of their unitary
representations allows us to determine the available spectrum of spin and
statistics on $M$ in a certain class of quantum theories. One interesting
result is that half-integral spin quantizations are obtained on certain
manifolds having an obstruction to an ordinary spin structure. We also compare
our results to corresponding ones for topological solitons in
$O(d+1)$-invariant nonlinear sigma models in $(d+1)$-dimensions, generalizing
recent studies in two spatial dimensions. Finally, we prove that there exists a
general scalar quantum theory yielding half-integral spin for particles (or
$O(d+1)$ solitons) on a closed, orientable manifold $M$ if and only if $M$
possesses a ${\rm spin}_c$ structure.
