The MNIST dataset is a large collection of handwritten digits. Each image is represented as a real-valued function on a 28x28 grid, and the task is to classify each image as the corresponding digit. This classification problem can be solved with high accuracy using classical machine learning (ML) methods. Garin and Tauzin applied Topological Data Analysis (TDA) to this dataset by interpreting each image as a cubical complex, introducing several filtrations, and analyzing the resulting persistent homology modules. Using different vectorization methods, the persistence diagrams obtained from these filtrations provide a compact set of features that can be used as input to ML algorithms. With this pipeline, Garin and Tauzin achieved accuracies comparable to classical methods while using only a small number of features. There is an alternative approach to MNIST classification due to Lalovic, who extracts a skeleton graph from each digit and computes persistent homology on a filtered simplicial complex derived from this graph. The aim of this talk is to give a survey of different approaches to MNIST classification using TDA. All necessary definitions will be given during the talk.