Linear-Gaussian dynamical systems (LDSs) are computationally tractable because all latents and observations are jointly Gaussian. However, these systems are too restrictive to satisfactorily model many dynamical systems of interest. One generalization, the switching linear dynamical system (SLDS), trades analytic tractability for a more expressive model, allowing a discrete set of different linear regimes to model the data. Here we introduce a switching linear dynamical system with a continuum of linear regimes that are traversed continuously in time. We call this model a \textit{continuous switching linear dynamical system} (CSLDS) and derive efficient variational Bayesian methods for inference and model learning.