Simple models of interacting spins have an important role in physics. They capture the properties of many magnetic materials, but also extend to other systems, such as bosons and fermions in a lattice, gauge theories, high-temperature superconductors, quantum spin liquids, and systems with exotic particles such as anyons and Majorana fermions(1,2). To study and compare these models, a versatile platform is needed. Realizing such systems has been a long-standing goal in the field of ultracold atoms. So far, spin transport has only been studied in systems with isotropic spin-spin interactions(3-12). Here we realize the Heisenberg model describing spins on a lattice, with fully adjustable anisotropy of the nearest-neighbour spin-spin couplings (called the XXZ model). In this model we study spin transport far from equilibrium after quantum quenches from imprinted spin-helix patterns. When spins are coupled only along two of three possible orientations (the XX model), we find ballistic behaviour of spin dynamics, whereas for isotropic interactions (the XXX model), we find diffusive behaviour. More generally, for positive anisotropies, the dynamics ranges from anomalous superdiffusion to subdiffusion, whereas for negative anisotropies, we observe a crossover in the time domain from ballistic to diffusive transport. This behaviour is in contrast with expectations from the linear-response regime and raises new questions in understanding quantum many-body dynamics far away from equilibrium.