The Toda lattice is a system of ordinary differential equations describing the motion of particles on a line connected via springs. This Toda lattice system is notable for its ability to be rewritten as a Lax equation, a certain differential equation of time-dependent linear operators. In this talk, we will introduce the Lax equation and describe its connection to the Korteweg-de Vries equation, as well as establish the Toda lattice in the form of a Lax equation. Time permitting, we will also discuss several methods for solving the Toda lattice system that use orthogonal polynomials, QR factorization, and a Riemann-Hilbert problem.