An astonishingly diverse biomolecular circuitry orchestrates the functioning machinery underlying every living cell. These biomolecules and their circuits have been engineered not only for various industrial applications but also to perform other atypical functions that they were not evolved for\textemdash including computation. Various kinds of computational challenges, such as solving NP-complete problems with many variables, logical computation, neural network operations, and cryptography, have all been attempted through this unconventional computing paradigm. In this review, we highlight key experiments across three different ``eras'' of molecular computation, beginning with molecular solutions, transitioning to logic circuits and ultimately, more complex molecular networks. We also discuss a variety of applications of molecular computation, from solving NP-hard problems to self-assembled nanostructures for delivering molecules, and provide a glimpse into the exciting potential that molecular computing holds for the future. Also see the video abstract here: https://youtu.be/9Mw0K0vCSQw