  We extend basic results in $3$-manifold topology to general three-dimensional
Alexandrov spaces (or Alexandrov $3$-spaces for short), providing a unified
framework for manifold and non-manifold spaces. We generalize the connected sum
to non-manifold $3$-spaces and prove a prime decomposition theorem, exhibit an
infinite family of closed, prime non-manifold $3$-spaces which are not
irreducible, and establish a conjecture of Mitsuishi and Yamaguchi on the
structure of closed, simply-connected Alexandrov $3$-spaces with non-negative
curvature. Additionally, we define a notion of generalized Dehn surgery for
Alexandrov $3$-spaces and show that any closed Alexandrov $3$-space may be
obtained by performing generalized Dehn surgery on a link in $S^3$ or the
non-trivial $S^2$-bundle over $S^1$. As an application of this result, we show
that every closed Alexandrov $3$-space is homeomorphic to the boundary of a
$4$-dimensional Alexandrov space.
