since 2005 a class of algebras, the {\it leavitt path algebras} $l_k(e)$ (for $k$ any field and $e$ any directed graph), has been a focus of investigation by both algebraists and c$^*$-analysts. in this talk i'll define these algebras, and give some of their general properties. then i'll describe some of the current lines of investigation in the area. in particular, i'll show a connection between ideas from symbolic dynamics (``flow equivalence") and the grothendieck group $k_0(l_k(e))$. with that connection in mind, i'll explain one of the most compelling open problems in leavitt path algebras, the {\it algebraic kirchberg phillips question}, which can be paraphrased as: can we recover $l_k(e)$ from $k_0(l_k(e))$? while the answer to the corresponding question for graph c$^*$-algebras is {\it yes}, there remains a barrier to a complete answer on the algebra side.