We adapt the classical definition of locally stationary processes in discrete-time to the continuous-time setting and obtain equivalent representations in the time and frequency domain. From this, a unique time-varying spectral density is derived using the Wigner-Ville spectrum. As an example, we investigate time-varying L\évy-driven state space processes, including the class of time-varying L\évy-driven CARMA processes. First, the connection between these two classes of processes is examined. Considering a sequence of time-varying L\évy-driven state space processes, we then give sufficient conditions on the coefficient functions that ensure local stationarity with respect to the given definition.
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