Variational　Auto-Encoder　(VAE)　is　an　important　probabilistic　technology　to　model　1D　vectorial　data.　However,　when　applying　VAE　model　to　2D　image,　vectorization　is　necessary.　Vectorization　process　may　lead　to　dimension　curse　and　lose　valuable　spatial　information.　To　avoid　these　problems,　we　propose　a　novel　VAE　model　based　on　matrix　variables　named　as　Matrix-variate　Variational　Auto-Encoder　(MVVAE).　In　this　model,　input,　hidden　and　latent　variables　are　all　in　matrix　form,　therefore　inherent　spatial　structure　of　2D　images　can　be　maintained　and　utilized　better.　Especially,　the　latent　variable　is　assumed　to　follow　matrix　Gaussian　distribution　which　is　more　suitable　for　describing　2D　images.　To　solve　the　weights　and　the　posterior　of　latent　variable,　the　variational　inference　process　is　given.　The　experiments　are　designed　for　three　real-world　application:　reconstruction,　denoising　and　completion.　The　experimental　results　demonstrate　that　MVVAE　shows　better　performance　than　VAE　and　other　probabilistic　methods　for　modeling　and　processing　2D　data.　(C)　2020　Elsevier　Inc.　All　rights　reserved.