In this paper, we consider a general class of vector error correction models which allow for asymmetric and non-linear error correction. We provide asymptotic results for (quasi-)maximum likelihood (QML) based estimators and tests. General hypothesis testing is considered, where testing for linearity is of particular interest as parameters of non-linear components vanish under the null. To solve the latter type of testing, we use the so-called sup tests, which here requires development of new (uniform) weak convergence results. These results are potentially useful in general for analysis of non-stationary non-linear time series models. Thus the paper provides a full asymptotic theory for estimators as well as standard and non-standard test statistics. The derived asymptotic results prove to be new compared to results found elsewhere in the literature due to the impact of the estimated cointegration relations. With respect to testing, this makes implementation of testing involved, and bootstrap versions of the tests are proposed in order to facilitate their usage. The asymptotic results regarding the QML estimators extend results in Kristensen and Rahbek (2010, Journal of Econometrics) where symmetric non-linear error correction are considered. A simulation study shows that the finite sample properties of the bootstrapped tests are satisfactory with good size and power properties for reasonable sample sizes.