  Irreducible representations of both Leavitt and Cohn path algebras of an
arbitrary digraph with coefficients in a commutative field is classified. They
are constructed in several ways using both infinite paths on the right as well
as direct limits or factors of one-sided projective ideals of the ordinary
quiver algebra, respectively. Furthermore, their defining relations are
described, too, whence criterions are easily given when they are finitely
presented or finite dimensional. Moreover, their endomorphism rings,
annihilator primitive ideals are also computed directly. In particular, the
cardinality of the set of sinks or infinite emitters, respectively, is an
invariant of Leavitt path algebras.
