This review sets out to discuss the foremost applications of artificial intelligence (AI), particularly deep learning (DL) algorithms, in single-photon emission computed tomography (SPECT) and positron emission tomography (PET) imaging. To this end, the underlying limitations/challenges of these imaging modalities are briefly discussed followed by a description of AI-based solutions proposed to address these challenges. This review will focus on mainstream generic fields, including instrumentation, image acquisition/formation, image reconstruction and low-dose/fast scanning, quantitative imaging, image interpretation (computer-aided detection/diagnosis/prognosis), as well as internal radiation dosimetry. A brief description of deep learning algorithms and the fundamental architectures used for these applications is also provided. Finally, the challenges, opportunities, and barriers to full-scale validation and adoption of AI-based solutions for improvement of image quality and quantitative accuracy of PET and SPECT images in the clinic are discussed.