Continuity equation:
A continuity equation in physics is a differential equation that describes the transport of some kind of conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved, a vast variety of physics may be described with continuity equations.
Continuity equations are the (stronger) local form of conservation laws. All the examples of continuity equations below express the same idea,

which is roughly that: the total amount (of the conserved quantity) inside any region can only change by the amount that passes in or out of the region through the boundary. A conserved quantity cannot increase or decrease, it can only move from place to place.

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".