This paper provides an introduction to variational autoencoders (VAEs) and their extensions. VAEs are a type of deep latent-variable model that combines probabilistic modeling with neural networks. The key idea is to approximate the true posterior distribution over latent variables using an encoder model, which is then used to train a generative model. The VAE framework allows for efficient optimization of the evidence lower bound (ELBO), which is a lower bound on the log-likelihood of the data. The ELBO is composed of two terms: the expected log-likelihood of the data given the latent variables and the Kullback-Leibler (KL) divergence between the approximate posterior and the true posterior. The VAE framework is particularly useful for learning deep latent-variable models and corresponding inference models. It allows for efficient optimization using stochastic gradient descent (SGD) and can be extended to handle more complex models. The paper discusses various aspects of VAEs, including the encoder and decoder models, the ELBO, the reparameterization trick, and the use of factorized Gaussian posteriors. It also covers related work and extensions, such as deeper generative models and alternative methods for increasing expressivity. The paper concludes with a discussion of the advantages and challenges of VAEs and their potential applications in various domains.This paper provides an introduction to variational autoencoders (VAEs) and their extensions. VAEs are a type of deep latent-variable model that combines probabilistic modeling with neural networks. The key idea is to approximate the true posterior distribution over latent variables using an encoder model, which is then used to train a generative model. The VAE framework allows for efficient optimization of the evidence lower bound (ELBO), which is a lower bound on the log-likelihood of the data. The ELBO is composed of two terms: the expected log-likelihood of the data given the latent variables and the Kullback-Leibler (KL) divergence between the approximate posterior and the true posterior. The VAE framework is particularly useful for learning deep latent-variable models and corresponding inference models. It allows for efficient optimization using stochastic gradient descent (SGD) and can be extended to handle more complex models. The paper discusses various aspects of VAEs, including the encoder and decoder models, the ELBO, the reparameterization trick, and the use of factorized Gaussian posteriors. It also covers related work and extensions, such as deeper generative models and alternative methods for increasing expressivity. The paper concludes with a discussion of the advantages and challenges of VAEs and their potential applications in various domains.