The rings studied by students in most rst-year algebra courses turn out to have what's known
as the "Invariant Basis Number" property: for every pair of positive integers m and n, if the free
left R-modules R^m and Rⁿ are isomorphic, then m = n. For instance, the IBN property in
the context of fields boils down to the statement that any two bases of a vector space must have
the same cardinality. Similarly, the IBN property for the ring of integers is a consequence of the
Fundamental Theorem for Finitely Generated Abelian Groups.

In seminal work completed the early 1960's, Bill Leavitt produced a specific, universal collection
of algebras which fail to have IBN. While it's fair to say that these algebras were initially viewed
as mere pathologies, it's just as fair to say that these now-so-called Leavitt algebras currently play
a central, fundamental role in numerous lines of research in both algebra and analysis.
More generally, from any directed graph E and any field K one can build the Leavitt path
algebra LK(E). In particular, the Leavitt algebras arise in this more general context as the algebras
corresponding to the graphs consisting of a single vertex. The Leavitt path algebras were first defined
in 2004; as of 2015 the subject has matured well into adolescence, currently enjoying a seemingly
constant opening of new lines of investigation, and the signicant advancement of existing lines.
I'll give an overview of some of the work on Leavitt path algebras which has occurred in their first
decade of existence, as well as mention some of the future directions and open questions in the
subject.

There should be something for everyone in this presentation, including and especially algebraists,
analysts, and graph theorists. We'll also present an elementary number theoretic observation which
provides the foundation for one of the recent main results in Leavitt path algebras, a result which
has had a number of important applications, including one in the theory of simple groups. The
talk will be aimed at a general audience; for most of the presentation, a basic course in rings and
modules will provide more-than-adequate background.