This work considers industrial process monitoring using a variational autoencoder (VAE). As a powerful deep generative model, the variational autoencoder and its variants have become popular for process monitoring. However, its monitoring ability, especially its fault diagnosis ability, has not been well investigated. In this paper, the process modeling and monitoring capabilities of several VAE variants are comprehensively studied. First, fault detection schemes are defined in three distinct ways, considering latent, residual, and the combined domains. Afterwards, to conduct the fault diagnosis, we first define the deep contribution plot, and then a deep reconstruction-based contribution diagram is proposed for deep domains under the fault propagation mechanism. In a case study, the performance of the process monitoring capability of four deep VAE models, namely, the static VAE model, the dynamic VAE model, and the recurrent VAE models (LSTM-VAE and GRU-VAE), has been comparatively evaluated on the industrial benchmark Tennessee Eastman process. Results show that recurrent VAEs with a deep reconstruction-based diagnosis mechanism are recommended for industrial process monitoring tasks.