  Full control over the dynamics of interacting, indistinguishable quantum
particles is an important prerequisite for the experimental study of strongly
correlated quantum matter and the implementation of high-fidelity quantum
information processing. Here we demonstrate such control over the quantum walk
- the quantum mechanical analogue of the classical random walk - in the strong
interaction regime. Using interacting bosonic atoms in an optical lattice, we
directly observe fundamental effects such as the emergence of correlations in
two-particle quantum walks, as well as strongly correlated Bloch oscillations
in tilted optical lattices. Our approach can be scaled to larger systems,
greatly extending the class of problems accessible via quantum walks
