Optical lattices provide a key enabling and controllable platform for exploring new physical phenomena and implications of degenerate quantum gases both in the quantum and nonlinear regimes. Based on the Gross- Pitaevskii/nonlinear Schrodinger equation with competing cubic-quintic nonlinearity, we show, numerically and theoretically, the nonlinear localization of dense Bose-Einstein condensates (BECs) in a novel two-dimensional twisted periodic potential called Moire optical lattices which, in essence, build a bridge between the perfect optical lattices and aperiodic ones. Our theory reveals that the Moire optical lattices display a wider second gap and flat-band feature, and support two kinds of localized matter-wave structures like gap solitons and topological states (gap vortices) with vortex charge s = 1, all populated inside the finite gaps of the linear Bloch-wave spectrum. We demonstrate, by means of linear-stability analysis and direct perturbed evolutions, that these localized structures have wide stability regions, paving the way for studying flat-band and Moire physics in shallow optical lattices and for finding robust coherent matter waves therein. The twisted periodic structures can be readily implemented with currently available optical-lattice technique in BECs and nonlinear optics experiments where the results predicted here are observable.