  This survey reports on current progress of programs to classify graph
C*-algebras and Leavitt path algebras up to Morita equivalence using K-theory.
Beginning with an overview and some history, we trace the development of the
classification of simple and nonsimple graph C*-algebras and state theorems
summarizing the current status of these efforts. We then discuss the much more
nascent efforts to classify Leavitt path algebras, and we describe the current
status of these efforts as well as outline current impediments that must be
solved for this classification program to progress. In particular, we give two
specific open problems that must be addressed in order to identify the correct
K-theoretic invariant for classification of simple Leavitt path algebras, and
we discuss the significance of various possible outcomes to these open
problems.
