  A free semigroupoid algebra is the closure of the algebra generated by a TCK
family of a graph in the weak operator topology. We obtain a structure theory
for these algebras analogous to that of free semigroup algebra. We clarify the
role of absolute continuity and wandering vectors. These results are applied to
obtain a Lebesgue-von Neumann-Wold decomposition of TCK families, along with
reflexivity, a Kaplansky density theorem and classification for free
semigroupoid algebras. Several classes of examples are discussed and developed,
including self-adjoint examples and a classification of atomic free
semigroupoid algebras up to unitary equivalence.
