This refereed collection of research papers and survey articles reflects the interplay of finite-dimensional algebras with other areas, such as algebraic geometry, homological algebra, and the theory of quantum groups. Modern trends are presented from the discussions at the AMS-IMS-SIAM Joint Summer Research Conference at the University of Washington, Seattle. The volume features several expository articles which aim to introduce inspiration to researchers in related areas, as it includes original papers spanning a broad spectrum of representation theory.