  We give a self-contained introduction to isolated points on curves and their
counterpoint, parameterized points, that situates these concepts within the
study of the arithmetic of curves. In particular, we show how natural geometric
constructions of infinitely many degree $d$ points on curves motivate the
definitions of $\mathbb{P}^1$- and AV-parameterized points and explain how a
result of Faltings implies that there are only finitely many isolated points on
any curve. We use parameterized points to deduce properties of the density
degree set and review how the minimum density degree relates to the gonality.
The paper includes several examples that illustrate the possible behaviors of
degree $d$ points.
