Image processing plays a crucial role in maximising diagnostic quality of positron emission tomography (PET) images. Recently, deep learning methods developed across many fields have shown tremendous potential when applied to medical image enhancement, resulting in a rich and rapidly advancing literature surrounding this subject. This review encapsulates methods for integrating deep learning into PET image reconstruction and post-processing for low-dose imaging and resolution enhancement. A brief introduction to conventional image processing techniques in PET is firstly presented. We then review methods which integrate deep learning into the image reconstruction framework as either deep learning-based regularisation or as a fully data-driven mapping from measured signal to images. Deep learning-based post-processing methods for low-dose imaging, temporal resolution enhancement and spatial resolution enhancement are also reviewed. Finally, the challenges associated with applying deep learning to enhance PET images in the clinical setting are discussed and future research directions to address these challenges are presented.