In this talk I will introduce the field of discrete harmonic analysis, beginning with the work of Bourgain and Stein on polynomial averaging operators. The simplest example of these operators is the discrete maximal function along the squares,

\[ \sup_N \left| \frac{1}{N} \sum_{n \leq N} f(x-n^2) \right|, \; \; \; f \in \ell^2 \]

which essentially controls pointwise convergence of the polynomial averages,

\[ \frac{1}{N} \sum_{n \leq N} T^{n^2} f,\]

where $f \in L^2(X)$ for some probability space $X$, where $T :X \to X$ is a measure preserving transformation.

I will then discuss current lines of inquiry in the field, beyond the purview of polynomial radon transforms (averaging operators or singular integrals along polynomial orbits).