Directed spaces, topological spaces equipped with some temporal structure, abound in nature. Examples include state spaces of complex systems, spacetimes, and classifying spaces of small categories. Algebraic invariants respecting the temporal structure on such spaces can tractably detect dynamic features of such spaces unseen by classical homotopy invariants on the underlying spaces. In this talk, I will introduce basic ideas, results, and calculations in directed algebraic topology and sketch some applications to the formal verification of software, graph theory, and combinatorial semigroup theory (time permitting).