In this paper we seek to estimate the state of a continuous-time dynamical process governed by a nonlinear stochastic differential equation, observed through discrete-time measurements. As the Bayesian posterior density is difficult to compute, we use variational inference (VI)-a method from machine learning-to approximate it. This is achieved by seeking the closest Gaussian density to the posterior, in the sense of the Kullback-Leibler divergence. The obtained algorithm, called the continuous-discrete variational Kalman filter (CD-VKF), provides implicit formulas that solve the considered problem. Our hope is that such a Kalman filter variant may prove more stable as it optimizes a closeness to the target distribution. We first clarify the connections between many Kalman filter variants and VI, then develop closed form approximate formulas for the CD-VKF. Our algorithm gives state of the art performances on the problem of reentry tracking of a space capsule.