  The numerical range of a bounded, linear operator on a Hilbert space is a set
in $\mathbb{C}$ that encodes important information about the operator. In this
survey paper, we first consider numerical ranges of matrices and discuss
several connections with envelopes of families of curves. We then turn to the
shift operator, perhaps the most important operator on the Hardy space
$H^2(\mathbb{D})$, and compressions of the shift operator to model spaces,
i.e.~spaces of the form $H^2 \ominus \theta H^2$ where $\theta$ is inner. For
these compressions of the shift operator, we provide a survey of results on the
connection between their numerical ranges and the numerical ranges of their
unitary dilations. We also discuss related results for compressed shift
operators on the bidisk associated to rational inner functions and conclude the
paper with a brief discussion of the Crouzeix conjecture.
