  In this article, we provide an overview of a one-to-one correspondence
between representations of the generalized Clifford algebra $C_f$ of a ternary
cubic form $f$ and certain vector bundles (called Ulrich bundles) on a cubic
surface $X$. We study general properties of Ulrich bundles, and using a recent
classification of Casanellas and Hartshorne, deduce the existence of
irreducible representations of $C_f$ of every possible dimension.
