Let C be an algebraic curve over Q, i.e., a 1-dimensional complex manifold defined by polynomial equations with rational coefficients. A celebrated result of Faltings implies that all algebraic points on C come in families of bounded degree, with finitely many exceptions. These exceptions are known as isolated points. We explore how these isolated points behave in families of curves and deduce consequences for the arithmetic of elliptic curves. We also explore how the beginnings of how the geometry of the parameterized points can be used to explore the arithmetic of the curve This talk is on joint work with A. Bourdon, Ö. Ejder, Y. Liu, and F. Odumodu, with I. Vogt, and with work in progress with I. Balçik, S. Chan, and Y. Liu.