  Latent Gaussian models (LGMs) are perhaps the most commonly used class of
models in statistical applications. Nevertheless, in areas ranging from
longitudinal studies in biostatistics to geostatistics, it is easy to find
datasets that contain inherently non-Gaussian features, such as sudden jumps or
spikes, that adversely affect the inferences and predictions made from an LGM.
These datasets require more general latent non-Gaussian models (LnGMs) that can
handle these non-Gaussian features automatically. However, fast implementation
and easy-to-use software are lacking, which prevent LnGMs from becoming widely
applicable. In this paper, we derive variational Bayes algorithms for fast and
scalable inference of LnGMs. The approximation leads to an LGM that downweights
extreme events in the latent process, reducing their impact and leading to more
robust inferences. It can be applied to a wide range of models, such as
autoregressive processes for time series, simultaneous autoregressive models
for areal data, and spatial Mat\'ern models. To facilitate Bayesian inference,
we introduce the ngvb package, where LGMs implemented in R-INLA can be easily
extended to LnGMs by adding a single line of code.
