This talk is about a theory of local newforms for irreducible, admissible representations of the group $GSp(4,F)$, where $F$ is a non-archimedean local field. We will present a joint work with Brooks Roberts, where we proved the existence of a local newform theory analogous to Casselman's theory for $GL(2,F)$. A key feature of our theory is to consider fixed vectors under a family of open-compact subgroups called the paramodular groups. Local newforms provide a link between classical modular forms and automorphic representations. Our theory therefore has applications to Siegel modular forms of degree $2$; this will be discussed in a talk by Brooks Roberts.