  We study the intersection theory of punctured pseudoholomorphic curves in
$4$-dimensional symplectic cobordisms. We first study the local intersection
properties of such curves at the punctures. We then use this to develop
topological controls on the intersection number of two curves. We also prove an
adjunction formula which gives a topological condition that will guarantee a
curve in a given homotopy class is embedded, extending previous work of
Hutchings. We then turn our attention to curves in the symplectization
$\mathbb{R}\times M$ of a $3$-manifold $M$ admitting a stable Hamiltonian
structure. We investigate controls on intersections of the projections of
curves to the $3$-manifold, and we present conditions that will guarantee the
projection of a curve to the $3$-manifold is an embedding. Finally we consider
an application concerning pseudoholomorphic curves in manifolds admitting a
certain class of holomorphic open book decomposition, and an application
concerning the existence of generalized pseudoholomorphic curves.
