In deep latent Gaussian models, the latent variable is generated by a\ntime-inhomogeneous Markov chain, where at each time step we pass the current\nstate through a parametric nonlinear map, such as a feedforward neural net, and\nadd a small independent Gaussian perturbation. This work considers the\ndiffusion limit of such models, where the number of layers tends to infinity,\nwhile the step size and the noise variance tend to zero. The limiting latent\nobject is an It\\^o diffusion process that solves a stochastic differential\nequation (SDE) whose drift and diffusion coefficient are implemented by neural\nnets. We develop a variational inference framework for these \\textit{neural\nSDEs} via stochastic automatic differentiation in Wiener space, where the\nvariational approximations to the posterior are obtained by Girsanov\n(mean-shift) transformation of the standard Wiener process and the computation\nof gradients is based on the theory of stochastic flows. This permits the use\nof black-box SDE solvers and automatic differentiation for end-to-end\ninference. Experimental results with synthetic data are provided.\n