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The aim of this review article is to present recent advances in the theory of the dynamics of modulated phases in the Frenkel-Kontorova model. This theory is motivated through two specific condensed matter systems: charge-density wave conductors and Josephson junction arrays. The presentation tries to integrate the existing results into the perspective of the equilibrium theory of the model, which is summarized in the beginning. The issues of defectibility, metastability, pinning and synchronization are discussed in connection with the underlying interplay of continuum and discrete descriptions. Special emphasis is placed on the different transitions between dynamical phases; namely depinning transition, unlocking transition and dynamical Aubry transition.