  Let L be the Leavitt path algebra of an arbitrary directed graph E over a
field K. This survey article describes how this highly non-commutative ring L
shares a number of the characterizing properties of a Dedekind domain or a
Pr\"ufer domain expressed in terms of their ideal lattices. Special types of
ideals such as the prime, the primary, the irreducible and the radical ideals
of L are described by means of the graphical properties of E. The existence and
the uniqueness of the factorization of a non-zero ideal of L as an irredundant
product of prime or primary or irreducible ideals is established. Such
factorization always exists for every ideal in L if the graph E is finite or if
L is two-sided artinian or two-sided noetherian. In all these factorizations,
the graded ideals of L seem to play an important role. Necessary and sufficient
conditions are given under which L is a generalized ZPI ring, that is, when
every ideal of L is a product of prime ideals. Intersections of various special
types of ideals are investigated and an anlogue of Krull's theorem on the
intersection of powers of an ideal in L is established.
