We describe a general technique that allows one to induce and control strong interaction between spin states of neighboring atoms in an optical lattice. We show that the properties of spin exchange interactions; such as magnitude; sign; and anisotropy; can be designed by adjusting the optical potentials. We illustrate how this technique can be used to efficiently “engineer” quantum spin systems with desired properties; for specific examples ranging from scalable quantum computation to probing a model with complex topological order that supports exotic anyonic excitations.