Recommender systems have become an important instrument to connect people to information. Sparse, complex, and rapidly growing data presents new challenges to traditional recommendation algorithms. To overcome these challenges, various deep learning-based recommendation algorithms have been proposed. Among these, Variational AutoEncoder (VAE)-based recommendation methods stand out. VAEs are based on a flexible probabilistic framework, which is robust for data sparsity and compatible with other deep learning-based models for dealing with multimodal data. In addition, the deep generative structure of VAEs helps to perform Bayesian inference in an efficient manner. VAE-based recommendation algorithms have given rise to many novel graphical models and they have achieved promising performance. In this paper, we conduct a survey to systematically summarize recent VAE-based recommendation algorithms. Four frequently used characteristics of VAE-based recommendation algorithms are summarized, and a taxonomy of VAE-based recommendation algorithms is proposed. We also identify future research directions for, advanced perspectives on, and the application of VAEs in recommendation algorithms, to inspire future work on VAEs for recommender systems.