Abstract: We provide an overview of results for evolution of discrete time quantum walks on lattices. The focus is on homogeneous walks where Fourier analysis is applicable, which allows to investigate the spectrum of the evolution operator in detail. Most of the features of the probability distribution generated by the quantum walk evolution, e.g. the characteristic peaks and ballistic spreading, are captured in the limit density, which can be derived from the convergence moments of position rescalled with the number of steps. The derivation of the limit density is illustrated on the example of the quantum walk on a line with the Hadamard coin. We then turn to the examples of quantum walks on a line and a plane with the Grover coin, where the evolution operator has a non-empty point spectrum. This results in the trapping effect, where the walker remains localized in the vicinity of the starting point with non-vanishing probability.