  In this paper we define and characterize cointegrated continuous-time linear
state-space models. A main result is that a cointegrated continuous-time linear
state-space model can be represented as a sum of a L\'evy process and a
stationary linear state-space model. Moreover, we prove that the class of
cointegrated multivariate L\'evy-driven autoregressive moving-average (MCARMA)
processes, the continuous-time analogues of the classical vector ARMA
processes, is equivalent to the class of cointegrated continuous-time linear
state space models. Necessary and sufficient conditions for MCARMA processes to
be cointegrated are given as well extending the results of Comte for MCAR
processes. The conditions depend on the autoregressive polynomial. Finally, we
investigate cointegrated continuous-time linear state-space models observed on
a discrete time-grid and derive an error correction form for this model. The
error correction form is based on an infinite linear filter in contrast to the
finite linear filter for VAR models.
