Respiratory and cardiac motion causes qualitative and quantitative inaccuracies in whole body multi-modality imaging such as positron emission tomography coupled with computed tomography (PET/CT) and positron emission tomography/magnetic resonance imaging (PET/MRI). Solutions presented to date include motion synchronized PET and corresponding anatomical acquisitions (four dimensional [4D] PET/CT, 4D PET/MR), frequently referred to as the gating approach. This method is based on the acquisition of an external surrogate using an external device (pressure belt, optical monitoring system, spirometer etc.), subsequently used to bin PET and CT or MR anatomical data into a number of gates. A first limitation of this method is the low signal to noise ratio (SNR) of the resulting motion synchronized PET frames, given that every reconstructed frame contains only part of the count statistics available throughout a motion average PET acquisition. Another limitation is that the complex motion of internal organs cannot be fully estimated, characterized and modelled using a mono-dimensional motion signal. In order to resolve such issues, many advanced techniques have been proposed which include three consecutive major steps. These are based on firstly acquiring an external or internal motion surrogate, estimating or modelling the internal motion using anatomical information extracted from 4D anatomical images (CT and/or MR) and finally correcting for motion either in the PET raw data space, the image space or incorporate it within the PET image reconstruction which is the most optimal based motion correction method in PET/CT and in PET/MR imaging. Current research efforts are concentrating on combining the last two steps within a joint motion estimation/motion correction approach, the exploitation of MRI specific motion characterization sequences and the combination of both respiratory and cardiac motion corrections. The goal of this review is to present and discuss the different steps of all these motion correction methods in PET/CT and PET/MR imaging for whole body applications.