A continuity equation is a differential equation that describes the conservative transport of some kind of quantity. Since mass, energy, momentum, and other natural quantities are conserved, a vast variety of physics may be described with continuity equations.
All the examples of continuity equations below express the same idea. Continuity equations are the (stronger) local form of conservation laws.
Any continuity equation has a "differential form" (in terms of the divergence operator) and an "integral form" (in terms of a flux integral). In this article, only the "differential form" versions will be given; see the article divergence theorem for how to express any of these laws in "integral form".
In fluid dynamics, the continuity equation is a mathematical statement that, in any steady state process, the rate at which mass enters a system is equal to the rate at which mass leaves the system. In fluid dynamics, the continuity equation is analogous to Kirchhoff's Current Law in electric circuits.