Suppose that $f : X \dashrightarrow X$ is a rational self-map of an algebraic variety, with everything defined over $\bar{\mathbb{Q}}$. If $P$ is a $\bar{\mathbb Q}$-point of $X$, then the arithmetic degree $\alpha_f(P)$ of Kawaguchi and Silverman is a measure of the rate of growth of the heights the points of $f^n(P)$, and is a sort of "arithmetic entropy" of $f$. In this talk, I will introduce higher arithmetic degrees, which measure the height growth of positive-dimensional cycles, and give some conjectures and results relating these to the dynamical degrees of $f$. This is joint with Nguyen-Bac Dang, Dragos Ghioca, Fei Hu, and Matthew Satriano.