Along with Markov chain Monte Carlo (mcmc) methods, variational inference (vi) has emerged as a central computational approach to large-scale Bayesian inference. Rather than sampling from the true posterior π, vi aims at producing a simple but effective approximation π to π for which summary statistics are easy to compute. However, unlike the well-studied mcmc methodology, algorithmic guarantees for vi are still relatively less well-understood. In this work, we propose principled methods for vi, in which π is taken to be a Gaussian or a mixture of Gaussians, which rest upon the theory of gradient flows on the Bures-Wasserstein space of Gaussian measures. Akin to mcmc, it comes with strong theoretical guarantees when π is log-concave.