Stomach cancer is the third deadliest type of cancer in the world (0.86 million deaths in 2017). In 2035, a 20% increase will be observed both in incidence and mortality due to demographic effects if no interventions are foreseen. Upper GI endoscopy (UGIE) plays a paramount role in early diagnosis and, therefore, improved survival rates. On the other hand, human and technical factors can contribute to misdiagnosis while performing UGIE. In this scenario, artificial intelligence (AI) has recently shown its potential in compensating for the pitfalls of UGIE, by leveraging deep learning architectures able to efficiently recognize endoscopic patterns from UGIE video data. This work presents a review of the current state-of-the-art algorithms in the application of AI to gastroscopy. It focuses specifically on the threefold tasks of assuring exam completeness (i.e., detecting the presence of blind spots) and assisting in the detection and characterization of clinical findings, both gastric precancerous conditions and neoplastic lesion changes. Early and promising results have already been obtained using well-known deep learning architectures for computer vision, but many algorithmic challenges remain in achieving the vision of AI-assisted UGIE. Future challenges in the roadmap for the effective integration of AI tools within the UGIE clinical practice are discussed, namely the adoption of more robust deep learning architectures and methods able to embed domain knowledge into image/video classifiers as well as the availability of large, annotated datasets.