  This survey of the recent developments in the investigations of a Leavitt
path algebra L of an arbitrary graph E over a field K consists of two parts. In
the first part describes how very often a single graph property of E implies
multiple ring properties of L, thus making Leavitt path algebras effective
tools in constructing rings of various desirable properties. The second part
describes methods of constructing simple modules over L and characterizes
Leavitt path algebras all of whose simple modules possess some specific
properties such as being flat, finitely presented,being graded, injective etc.
