\indent Diophantine equations, one of the oldest topics of mathematical research, remains the object of intense and fruitful study. A rational solution to a system of algebraic equations is tantamount to a point with rational coordinates (briefly, a "rational point") on the corresponding algebraic variety $V$. Already for $V$ of dimension 1 (an "algebraic curve"), many natural theoretical and computational questions remain open, especially when the genus $g$ of $V$ exceeds 1. (The genus is a natural measure of the complexity of $V$; for example, if $P$ is a nonconstant polynomial without repeated roots then the equation $y^2 = P(x)$ gives a curve of genus $g$ iff $P$ has degree $2g+1$ or $2g+2$.) Faltings famously proved that if $g>1$ then the set of rational points is finite (Mordell's conjecture), but left open the question of how its size can vary with $V$, even for fixed $g$. Even for $g=2$ there are curves with literally hundreds of points; is the number unbounded? We briefly review the structure of rational points on curves of genus 0 and 1, and then report on relevant work since Faltings on points on curves of genus $g > 1$.