We describe a theory of "multi-structures", and explore logics and languages that are natural for the study of these mathematical objects. The text is written for upper level undergraduate students and beginning graduate students in Computer Science, Computer Engineering, Mathematics, and Philosophy, although it is expected that students of other disciplines can benefit from the study of this subject as well. Multi-structures differ from the "structures" of classical logic and model theory in that the arity of a fundamental operation of a multi-structure is an ordered pair of nonnegative integers, such that the given operation is a function which maps "vectors" over the structure to other "vectors" over the same structure.