We introduce a new class of spherical colloids that reversibly self-assemble into well-defined nonlinear structures by virtue of "magnetic patches". This assembly is driven by tunable magnetostatic binding forces that originate from microscopic permanent magnets embedded underneath the surface of the particles. The resulting clusters form spontaneously in the absence of external magnetizing fields, and their geometry is determined by an interplay between magnetic, steric, and electrostatic interactions. Imposing an external magnetic field enables the clusters to unbind or change their geometry allowing, in. principle, the creation of materials with a reconfigurable structural arrangement.