We review both theoretical and experimental advances in the recently emerged field of modulated photonic lattices. These artificial periodic dielectric structures provide a powerful tool for the control of the fundamental aspects of light propagation. Photonic lattices are arrays of coupled optical waveguides, where the light propagation becomes effectively discretized. The discretized nature of light propagation gives rise to many new phenomena which are not possible in homogeneous bulk media, such as discrete diffraction and diffraction management, discrete and gap solitons, and discrete surface waves. Photonic lattices also allow one to realize optical analogies of phenomena occurring in other physical contexts, such as the physics of solid state and electron theory. For example, the light propagation in photonic lattices may resemble the motion of electrons in a crystalline lattice of semiconductor materials. Additionally, periodic modulation of a photonic lattice by varying its geometry or refractive index is analogous to applying a bias to control the motion of electrons in a crystalline lattice. An interplay between periodicity and modulation in photonic lattices opens up unique opportunities for tailoring diffraction and dispersion properties of light as well as controlling nonlinear interactions. First, we review the linear effects in the modulated waveguides and waveguide arrays, including optical Bloch oscillations and optical dynamic localization, that are key to the understanding of the modulation-driven diffraction management of light. Then we analyze the effects of array boundaries and defects, and highlight a new type of modulation-induced light localization based on the defect-free surface waves. Finally, we discuss nonlinear properties of the modulated lattices with an emphasis on their great potential for all-optical beam shaping and switching.