In this paper we consider cointegrated I (1) processes in the state-space framework. We introduce the state-space error correction model (SSECM) and discuss in detail how to estimate SSECMs by (pseudo) maximum likelihood methods, including reduced rank regression techniques which allow for a successive reduction of the number of parameters in the original constrained likelihood optimization problem. In doing so, we follow very closely the Johansen approach for the VAR case; see Johansen (1995). Finally, the remaining free parameters will be represented using a local parametrization technique and we show how efficient gradient-type algorithms can be employed for the numerical optimization in the resulting lower dimensional unconstrained problem. Simulation studies and applications are presented in a separate 'companion paper'; see Ribarits and Hanzon (2014).