Classical reduced order models rely on linear representations by reduced bases. Their performance is theoretically evaluated in comparison to the best possible linear methods, determined by the Kolmogorov $n$-width. More recently, focus has shifted towards problems, which do not allow effective linear representations, reflected in a poor Kolmogorov $n$-width. These are addressed by new techniques based on nonlinear representations. Some examples are transported subspaces, optimal transport and neural networks for parametric hyperbolic PDEs. In the first part of the talk, we present new reduced order models based on neural networks together with low rank weights, compression and a careful split into pre-trained and fine tuned components for online and offline phases. The second part of the talk considers performance benchmarks. The Kolmogorov $n$ width only lower bounds linear methods and is therefore unsuitable for modern nonlinear techniques. There are multiple generalizations suited for specific types of nonlinearities, e.g. library width provide benchmarks for dictionary methods and the $(M,N)$-width for transported subspaces. However, these width do not provide lower bounds for arbitrary nonlinear methods. This is be done by manifold width, stable width and entropy, which are typically too powerful, resulting in lower bounds of size zero. In this talk, we show that a shift in perspective allows us to use the latter width to obtain non-trivial lower bounds for all nonlinear methods, subject only to basic stability requirements.