High noise and low spatial resolution are two key confounding factors that limit the qualitative and quantitative accuracy of PET images. AI models for image denoising and deblurring are becoming increasingly popular for post-reconstruction enhancement of PET images. We present here a detailed review of recent efforts for AI-based PET image enhancement with a focus on network architectures, data types, loss functions, and evaluation metrics. We also highlight emerging areas in this field that are quickly gaining popularity, identify barriers to large-scale adoption of AI models for PET image enhancement, and discuss future directions.