A quantum particle moving on a line can show remarkable properties, completely different from classical particles. While in the well-known random walk at each time step a walker moves randomly one step to the right or to the left, a quantum particle can be brought into a coherent superposition of right and left. After several steps of such a quantum walk, the probability to find the particle at a certain position is dominated by matter wave interference of different partial wave packets of the walker. A prominent and characteristic consequence is the linear scaling of the width of the distribution with the number of time steps in the quantum case, in contrast to the diffusive scaling of the random walk. This linear scaling forms the basis for applications in quantum information science, such as fast searching algorithms. \par I will report on the experimental realization of a quantum walk using single neutral atoms trapped in a state-dependent optical lattice. Site-resolved fluorescence imaging in the lattice allows us to probe and characterize the delocalized wave function of an atom with local quantum state tomography. Further, we can perform an effective time-reversal and refocus a delocalized atom back onto its initial position. This reflects the highly coherent and deterministic nature of the quantum walk. In contrast, by destroying the coherence after each time step, we recover the classical random walk. \par Furthermore, we have developed novel tools to control the quantum motion of atoms in a state-dependent potential. I will report on a ground state cooling method base don microwaves and the ability to engineer arbitrary motional states.