  The subject of this paper is the bounded level curves of a meromorphic
function $f$ with domain $G$ such that each component of $\partial{G}$ consists
of a level curve of $f$. (A primary example of such a function being a ratio of
finite Blaschke products of different degrees, with domain $\mathbb{D}$.) We
will first prove several facts about a single bounded level curve of a $f$ in
isolation from the other level curves of $f$. We will then study how the level
curves of $f$ lie with respect to each other. It is natural to expect that the
sets $\{z:|f(z)|=\epsilon\}$ vary continuously as $\epsilon$ varies. We will
make this notion explicit, and use this continuity to prove several results
about the bounded level curves of $f$. It is well known that if $z_0$ is a zero
or a pole of $f$, then $f$ is conformally equivalent to the function
$z\mapsto{z^k}$ (for some $k\in\mathbb{Z}$) in a neighborhood of $z_0$. We
generalize this fact by finding a natural decomposition of $G$ into finitely
many sub-regions (also bounded by level curves of $f$), on each of which $f$ is
conformally equivalent to $z\mapsto{z^k}$ (for some $k\in\mathbb{Z}$). Also
included is a new proof, using level curves, of the Gauss--Lucas theorem that
the critical points of a polynomial are contained in the convex hull of the
polynomial's zeros.
