Moiré lattices are a universal and ubiquitous phenomenon. They result from the superposition of two identical periodic structures rotated with respect to each other. Such patterns appear in many different situations and physical systems, as well as in the everyday life. At the same time, moiré lattices are a profound concept that has emerged as a powerful tool for creation of material heterostructures that exhibit a variety of new phenomena with far-reaching implications, whose exploration is of paramount importance from fundamental and applied viewpoints. Moiré lattices exhibit properties that sharply contrast with those of their constituent periodic components. In this presentation I will describe our results on optically-induced moiré patterns in photorefractive crystals, where by changing only one parameter, such as the twist angle between two sublattices forming the pattern, one can explore the transition between structures with fundamentally different periodicity: periodic, general aperiodic, and quasi-crystal ones. I will discuss different scenarios of linear light propagation in optically induced moiré lattices, when the wavepacket can diffract or localize depending on the rotation angle and depths of two sublattices forming the structure. Two different scenarios of localization-delocalization transition will be illustrated, with one of them occurring upon variation of the depth of one of the sublattices, while in the other case such transition occurs upon variation of the rotation angle between the sublattices. I will also show how periodicity of moiré lattices affects propagation of nonlinear excitations and demonstrate sharp variations in thresholds for soliton formation controlled by the twist angle.