We introduce a technique that makes every explicit Runge--Kutta (ERK) time stepping method invariant-domain preserving and mass conservative when applied to high-order discretizations of the Cauchy problem associated with systems of nonlinear conservation equations. The key idea is that, at each stage of the ERK scheme, one computes a low-order update, a high-order update (both defined from the same intermediate stage using an incremental representation of the Butcher tableau), and then one applies a nonlinear mass conservative limiting operator. The main advantage over to the strong stability preserving (SSP) paradigm is more flexibility in the choice of the ERK scheme, thus allowing for a less stringent restriction on the time step. The technique is agnostic to the space discretization. It can be combined with continuous finite elements, discontinuous finite elements, finite volume, and finite difference discretizations in space. Numerical experiments are presented to illustrate the theory. In particular, we show second-order and third-order ERK schemes that outperform their SSP counterparts. We also show fifth-order ERK methods that are invariant-domain preserving, which is not possible within the SSP paradigm.