Positron emission tomography (PET) is vital for diagnosing diseases and monitoring treatments. Conventional image reconstruction (IR) techniques like filtered backprojection and iterative algorithms are powerful but face limitations. PET IR can be seen as an image-to-image translation. Artificial intelligence (AI) and deep learning (DL) using multilayer neural networks enable a new approach to this computer vision task. This review aims to provide mutual understanding for nuclear medicine professionals and AI researchers. We outline fundamentals of PET imaging as well as state-of-the-art in AI-based PET IR with its typical algorithms and DL architectures. Advances improve resolution and contrast recovery, reduce noise, and remove artifacts via inferred attenuation and scatter correction, sinogram inpainting, denoising, and super-resolution refinement. Kernel-priors support list-mode reconstruction, motion correction, and parametric imaging. Hybrid approaches combine AI with conventional IR. Challenges of AI-assisted PET IR include availability of training data, cross-scanner compatibility, and the risk of hallucinated lesions. The need for rigorous evaluations, including quantitative phantom validation and visual comparison of diagnostic accuracy against conventional IR, is highlighted along with regulatory issues. First approved AI-based applications are clinically available, and its impact is foreseeable. Emerging trends, such as the integration of multimodal imaging and the use of data from previous imaging visits, highlight future potentials. Continued collaborative research promises significant improvements in image quality, quantitative accuracy, and diagnostic performance, ultimately leading to the integration of AI-based IR into routine PET imaging protocols.