\( \newcommand{\bu}{\mathbf{u}} \newcommand{\bx}{\mathbf{x}} \newcommand{\bz}{\mathbf{z}} \newcommand{\bL}{\mathbf{L}} \) In this review, we will consider dynamical systems of the form % \begin{align} \frac{d}{dt}\bx(t) = \mathbf{f}(\bx(t)),\label{Eq:ContinuousDynamicsGeneral} \end{align} % where $\bx\in\mathcal{X}\subseteq\mathbb{R}^n$ is the state of the system and $\mathbf{f}$ is a vector field describing the dynamics. In general, the dynamics may also depend on time $t$, parameters $\boldsymbol{\beta}$, and external actuation or control $\bu(t)$. Although we omit these here for simplicity, they will be considered in later sections. A major goal of modern dynamical systems is to find a new vector of coordinates $\bz$ such that either \begin{equation} \bx=\boldsymbol{\varphi}(\bz) \quad\text{ or }\quad \bz=\boldsymbol{\varphi}(\bx) \label{eq:linearizing-coordinates} \end{equation} where the dynamics are simplified, or ideally, linearized: \begin{align} \frac{d}{dt}\bz = \bL \bz. \end{align} While in geometric dynamics, one asks for homeomorphic Koopman's 1931 paper was central to the celebrated proofs of the ergodic theorem by von Neumann~\cite{neumann1932pnas} and Birkhoff~\cite{birkhoff1931pnas,birkhoff1932pnas}. The history of these developments is fraught with intrigue, as discussed by Moore~\cite{moore2015pnas}.