Carlsten, B.E. JACoW Publishing Geneva, Switzerland This talk will review the physics of the phase-space manipulations in general, and emittance compensation specifically. All the key elements of the several schemes will be discussed to overview the broad topic of transverse phase-space manipulations, focusing on the underlying beam physics. The special schemes of non-symplectic manipulations will be discussed and the impact of the techniques for developed for electrons will be discussed for broader application as for ion beams.