We find a new family of exact solutions in membrane theory, representing toroidal membranes spinning in several planes. They have energy square proportional to the sum of the different angular momenta, generalizing Regge-type string solutions to membrane theory. By compactifying the eleven-dimensional theory on a circle and on a torus, we identify a family of new nonperturbative states of type IIA and type IIB superstring theory (which contains the perturbative spinning string solutions of type II string theory as a particular case). The solution represents a spinning bound state of D branes and fundamental strings. Then we find similar solutions for membranes on AdS₇ × S⁴ and AdS₄ × S⁷. We also consider the analogous solutions in SU (N) Matrix-theory, and compute the energy. They can be interpreted as rotating open strings with D0 branes attached to their endpoints.