There has been a longstanding interest in positron emission tomography (PET) in combination with computed tomography (CT). Mostly because of the lack of structural information in PET which makes it difficult to assess the precise location of tissue with metabolic uptake, whereas CT can provide impressive anatomical details. PET/CT designs are facing many challenges such as the conversion of CT numbers to attenuation coefficients, giving rise to artefacts due to the presence of high Zeff material. Patient motion during scans degrades image quality and subsequent analysis, and is a challenge especially as spatial resolution improves. Software based image fusion remains a complex issue outside the brain. State of the art image quality in a modern PET/CT system includes incorporation of point spread function (PSF) and time-of-flight (TOF) information into the reconstruction leading to the high resolution of today's PET/CT systems. This review outlines the background and current knowledge of the PET/CT system design, motion correction and reconstruction approaches.