This talk begins with an introduction to the Einstein equations in general relativity. We will then explore various hyperbolic formulations, in which the extra four degrees of freedom are carefully handled. In particular, two representative treatments will be presented: the "wave gauge" (or "wave/harmonic coordinates") on which Yvonne Choquet-Bruhat's pioneering 1952 local well-posedness result was based; and the "maximal time foliation" which plays an essential role in the proof of global stability of Minkowski spacetime by Christodoulou and Klainerman (1993).