  These are lecture notes of a minicourse given by the first author at the
Summer School on Quantization at the University of Notre Dame in June 2011. The
notes were written up and expanded by the second author who took the liberty of
adding a few interesting results and proofs from the literature. In a broad
sense, our goal is to give an introduction to representation theory of rational
Cherednik algebras and some of its recent applications. More specifically, we
focus on the two concepts featuring in the title (Dunkl operators and
quasi-invariants) and explain the relation between them. The course was
originally designed for graduate students and nonexperts in representation
theory. In these notes, we tried to preserve an informal style, even at the
expense of making imprecise claims and sacrificing rigor.
