$$\DeclareMathOperator\supp{supp}
\DeclareMathOperator*{\argmax}{arg,max}$$
The Variational Autoencoder (VAE) is a not-so-new-anymore Latent Variable Model (Kingma & Welling, 2014), which by introducing a probabilistic interpretation of autoencoders, allows to not only estimate the variance/uncertainty in the predictions, but also to inject domain knowledge through the use of informative priors, and possibly to make the latent space more interpretable. VAEs have various applications:
density estimation and sampling from the estimated density, i.e., data generation (e.g., image/sound/text generation and missing data imputation)[^1]
semi-supervised learning
representation learning for downstream tasks
VAEs are generative models