  In this paper we prove that a complete, embedded minimal surface $M$ in
$\mathbb{R}^3$ with finite topology and compact boundary (possibly empty) is
conformally a compact Riemann surface $\overline{M}$ with boundary punctured in
a finite number of interior points and that $M$ can be represented in terms of
meromorphic data on its conformal completion $\overline{M}$. In particular, we
demonstrate that $M$ is a minimal surface of finite type and describe how this
property permits a classification of the asymptotic behavior of $M$.
