Vector symbolic architectures (VSAs) are high-dimensional vector representations of objects (e.g., words, image parts), relations (e.g., sentence structures), and sequences for use with machine learning algorithms. They consist of a vector addition operator for representing a collection of unordered objects, a binding operator for associating groups of objects, and a methodology for encoding complex structures. We first develop constraints that machine learning imposes on VSAs; for example, similar structures must be represented by similar vectors. The constraints suggest that current VSAs should represent phrases (“ The smart Brazilian girl ”) by binding sums of terms, in addition to simply binding the terms directly. We show that matrix multiplication can be used as the binding operator for a VSA, and that matrix elements can be chosen at random. A consequence for living systems is that binding is mathematically possible without the need to specify, in advance, precise neuron-to-neuron connection properties for large numbers of synapses. A VSA that incorporates these ideas, Matrix Binding of Additive Terms (MBAT), is described that satisfies all constraints. With respect to machine learning, for some types of problems appropriate VSA representations permit us to prove learnability rather than relying on simulations . We also propose dividing machine (and neural) learning and representation into three stages, with differing roles for learning in each stage. For neural modeling, we give representational reasons for nervous systems to have many recurrent connections, as well as for the importance of phrases in language processing. Sizing simulations and analyses suggest that VSAs in general, and MBAT in particular, are ready for real-world applications.