By taking neural networks back to the school bench and teaching them some elements of geometry and topology we can build algorithms that can reason about the shape of data. Surprisingly these methods can be useful not only for computer vision – to model input data such as images or point clouds through global, robust properties – but in a wide range of applications, such as evaluating and improving the learning of embeddings, or the distribution of samples originating from generative models. This is the promise of the emerging field of Topological Data Analysis (TDA) which we will introduce and review recent works at its intersection with machine learning. TDA can be seen as being part of the increasingly popular movement of Geometric Deep Learning which encourages us to go beyond seeing data only as vectors in Euclidean spaces and instead consider machine learning algorithms that encode other geometric priors. In the past couple of years TDA has started to take a step out of the academic bubble, to a large extent thanks to powerful Python libraries written as extensions to scikit-learn or PyTorch.