Zero-shot recognition is centered around learning representations to transfer knowledge from seen to unseen classes. Where foundational approaches perform the transfer with semantic embedding spaces, e.g., from attributes or word vectors, the current state-of-the-art relies on prompting pre-trained vision-language models to obtain class embeddings. Whether zero-shot learning is performed with attributes, CLIP, or something else, current approaches de facto assume that there is a pre-defined embedding space in which seen and unseen classes can be positioned. Our work is concerned with real-world zero-shot settings where a pre-defined embedding space can no longer be assumed. This is natural in domains such as biology and medicine, where class names are not common English words, rendering vision-language models useless; or neuroscience, where class relations are only given with non-semantic human comparison scores. We find that there is one data structure enabling zero-shot learning in both standard and non-standard settings: a similarity matrix spanning the seen and unseen classes. We introduce four similarity-based zero-shot learning challenges, tackling open-ended scenarios such as learning with uncommon class names, learning from multiple partial sources, and learning with missing knowledge. As the first step for zero-shot learning beyond a pre-defined semantic embedding space, we propose κ-MDS, a general approach that obtains a prototype for each class on any manifold from similarities alone, even when part of the similarities are missing. Our approach can be plugged into any standard, hyperspherical, or hyperbolic zero-shot learner. Experiments on existing datasets and the new benchmarks show the promise and challenges of similarity-based zero-shot learning.