In problems such as transport-dominated systems, where the Kolmogorov $n$-widths decay slowly, traditional linear-subspace model order reduction (MOR) methods often fall short of achieving high-fidelity approximations. This challenge has driven the emergence of nonlinear MOR techniques, frequently powered by machine learning (ML) tools like autoencoders. In this talk, we present a unifying framework that bridges these nonlinear approaches with insights from differential geometry, offering a fresh perspective on MOR. By constructing embeddings for low-dimensional submanifolds and designing compatible reduction maps, we extend the classical (Petrov-)Galerkin framework to the manifold setting, providing a broader theoretical foundation for nonlinear MOR methods encompassing many methods from the literature. Moreover, the manifold setting provides a clear path to structure-preservation. Once the theoretical framework is available, the next step is to reveal practical lessons the general approach offers for specific methods. As a first example, we discuss the training of autoencoders in the context of MOR. While autoencoders typically require a nonlinear encoder-decoder pair and extensive hyperparameter tuning, we propose a method that extends the training data to enable a linear encoder without sacrificing accuracy.