  Understanding when an abstract complex curve of given genus comes equipped
with a map of fixed degree to a projective space of fixed dimension is a
foundational question; and Brill--Noether theory addresses this question via
linear series, which algebraically codify maps to projective targets. Classical
Brill--Noether theory, which focuses on smooth curves, has been intensively
explored; but much less is known for singular curves, particularly for those
with non-nodal singularities. In a one-parameter family of smooth curves
specializing to a singular curve $C_0$, one expects certain aspects of the
global geometry of the smooth fibers to ``specialize" to the local geometry of
the singularities of $C_0$. Making this expectation quantitatively precise
involves analyzing the arithmetic and combinatorics of semigroups ${\rm S}$
attached to discrete valuations defined on (the local rings of) these
singularities. In this largely-expository note we focus primarily on
Brill--Noether-type results for curves with {\it cusps}, i.e., unibranch
singularities; in this setting, the associated semigroups are {\it numerical}
semigroups with finite complement in $\mathbb{N}$.
