Abstract: Context. The connection between active galactic nuclei (AGN) and their host galaxies has been widely studied and found to be of great importance for providing answers to some fundamental questions related to AGN fuelling mechanisms, and both their formation and evolution. Aims. Using X-ray data and one of the deepest broad-band optical data sets available, we study how morphology and colours are related to X-ray properties for sources at redshifts z ? 2.0, using a sample of 262 AGN in the Subaru/XMM-Newton Deep Survey (SXDS). Methods. We performed our morphological classification using the galSVM code, which is a new method that is particularly suited to dealing with high-redshift sources. Colour?magnitude diagrams were studied in relationship to redshift, morphology, X-ray obscuration, and X-ray-to-optical flux ratio. We analysed the different regions in the colour?magnitude diagrams, and searched for correlations with the observed properties of AGN populations using models of their formation and evolution. Results. We confirm that a robust and reliable morphological classification of a general galaxy population at high redshift should be based on a multi-parametric approach. At least 50% of X-ray detected AGN at z ? 2.0 analysed in this work reside in spheroidal and bulge-dominated galaxies, while at least 18% have disk-dominated hosts. This suggests that different mechanisms may be responsible for triggering the nuclear activity. When analysing populations of X-ray detected AGN in both colour?magnitude and colour-stellar mass diagrams, the highest number of sources is found to reside in the green valley at redshifts ?0.5?1.5. However, a larger number of low-luminosity AGN have been detected than in previous works owing to the substantial depth of the SXDS optical data. Whether AGN are hosted by early- or late-type galaxies, no clear relationship has been found with the optical colours (independently of redshift), as is typical of normal galaxies. Both early- and late-type AGN cover similar ranges of X-ray obscuration, for both unobscured and obscured sources. Conclusions. Our findings appear to confirm some previous suggestions that X-ray selected AGN residing in the green valley represent a transitional population, quenching star formation by means of different AGN feedback mechanisms and evolving to red-sequence galaxies. They might be hosted by similar sources (the majority of sources being late-type elliptical and lenticular galaxies, and early-type spirals) with similar stellar populations, which are triggered mainly by major and/or minor mergers, and in some cases by means of secular mechanism, as shown in previous numerical simulations. In the aforementioned transition we observe different phases of AGN activity, with some AGN being in the ?QSO-mode? detected as compact, blue, and unobscured in X-rays, and with others passing through different phases before and after the ?QSO-mode?, being obscured and unobscured in X-rays, respectively

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