In this note we give an overview of various quantities that are used to measure the complexity of an algebraic dynamical system \(f:X\rightarrow {X}\) , including the dynamical degree \(\delta (f)\) , which gives a coarse measure of the geometric complexity of the iterates of f, the arithmetic degree \(\alpha (f,P)\) , which gives a coarse measure of the arithmetic complexity of the orbit of \(P\in {X}(\overline{\mathbb {Q}})\) , and various versions of the canonical height  \(\hat{h}_f(P)\) that provide more refined measures of arithmetic complexity. Emphasis is placed on open problems and directions for further exploration. ThisArithmetic dynamicsArithmetic complexity article is a slightly expanded version of a talk presented at the Simons Symposium on Algebraic, Complex and Arithmetic Dynamics held May 19–23, 2019 at Schloss Elmau, Krün, Germany. A small number of updates were added in 2023 and 2024 and are noted as such.