  We survey results in algebraic complexity theory, focusing on matrix
multiplication. Our goals are
  (i.) to show how open questions in algebraic complexity theory are naturally
posed as questions in geometry and representation theory, (ii.) to motivate
researchers to work on these questions, and (iii.) to point out relations with
more general problems in geometry. The key geometric objects for our study are
the secant varieties of Segre varieties. We explain how these varieties are
also useful for algebraic statistics, the study of phylogenetic invariants, and
quantum computing.
