  In this article, we classify (non-compact) $3$-manifolds with uniformly
positive scalar curvature. Precisely, we show that an oriented $3$-manifold has
a complete metric with uniformly positive scalar curvature if and only if it is
homeomorphic to an (possibly) infinite connected sum of spherical $3$-manifolds
and some copies of $\mathbb{S}^1\times \mathbb{S}^2$. Further, we study an
oriented $3$-manifold with mean convex boundary and with uniformly positive
scalar curvature. If the boundary is a disjoint union of closed surfaces, then
the manifold is an (possibly) infinite conned sum of spherical $3$-manifolds,
some handlebodies and some copies of $\mathbb{S}^1\times \mathbb{S}^2$.
