This tutorial introduces Variational Autoencoders (VAEs), a popular approach to unsupervised learning of complex distributions. VAEs are appealing because they are built on standard function approximators (neural networks) and can be trained using stochastic gradient descent. The tutorial covers the intuition behind VAEs, the mathematical details, and empirical behavior. It begins by discussing the challenges of generative modeling, particularly in high-dimensional spaces like images, and how VAEs address these challenges. The tutorial then delves into the mathematical foundation of VAEs, explaining the role of latent variables and the variational lower bound. It also covers the optimization process, including the reparameterization trick, which allows for efficient gradient descent. The tutorial further explores the information-theoretic interpretation of VAEs and their connection to minimum description length principles. Additionally, it discusses conditional VAEs, which can handle one-to-many input-to-output mappings, and provides examples of VAEs trained on datasets like MNIST. The tutorial concludes with a proof that VAEs have zero approximation error given arbitrarily powerful learners in one dimension.This tutorial introduces Variational Autoencoders (VAEs), a popular approach to unsupervised learning of complex distributions. VAEs are appealing because they are built on standard function approximators (neural networks) and can be trained using stochastic gradient descent. The tutorial covers the intuition behind VAEs, the mathematical details, and empirical behavior. It begins by discussing the challenges of generative modeling, particularly in high-dimensional spaces like images, and how VAEs address these challenges. The tutorial then delves into the mathematical foundation of VAEs, explaining the role of latent variables and the variational lower bound. It also covers the optimization process, including the reparameterization trick, which allows for efficient gradient descent. The tutorial further explores the information-theoretic interpretation of VAEs and their connection to minimum description length principles. Additionally, it discusses conditional VAEs, which can handle one-to-many input-to-output mappings, and provides examples of VAEs trained on datasets like MNIST. The tutorial concludes with a proof that VAEs have zero approximation error given arbitrarily powerful learners in one dimension.