  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\'evy-driven state space processes,
including the class of time-varying L\'evy-driven CARMA processes. First, the
connection between these two classes of processes is examined. Considering a
sequence of time-varying L\'evy-driven state space processes, we then give
sufficient conditions on the coefficient functions that ensure local
stationarity with respect to the given definition.
