Given a "data manifold" M⊂ℝ^n and "latent space" ℝ^ℓ, an autoencoder is a pair of continuous maps consisting of an "encoder" Eℝ^n→ℝ^ℓ and "decoder" Dℝ^ℓ→ℝ^n such that the "round trip" map D∘ E is as close as possible to the identity map _M on M. We present various topological limitations and capabilites inherent to the search for an autoencoder, and describe capabilities for autoencoding dynamical systems having M as an invariant manifold.