  We consider the problem of determining the class of continuous-time dynamical
systems that can be globally linearized in the sense of admitting an embedding
into a linear system on a finite-dimensional Euclidean space. We solve this
problem for dynamical systems having either a compact state space or at least
one compact attractor, obtaining necessary and sufficient conditions for the
existence of linearizing $C^k$ embeddings for $k\in \mathbb{N}_{\geq 0}\cup
\{\infty\}$. Corollaries include (i) several checkable necessary conditions for
global linearizability and (ii) extensions of the Hartman-Grobman and Floquet
normal form theorems beyond the classical settings. Overall, our results reveal
relationships between linearizability, symmetry, topology, and invariant
manifold theory. In particular, these relationships yield fundamental
limitations and capabilities of algorithms from the "applied Koopman operator
theory" literature.
