Measuring vital signs can reveal the state of body functioning and help to detect a health problem. In the state-of-the-art, numerous methods and devices are available for measuring vital signs. However, with the advent of artificial intelligence, new methods have been proposed that employ this technology. This paper aims to highlight the recent methods and devices based on artificial intelligence and novel techniques for measuring vital signs and processing algorithms. We analyzed 122 papers and classified them into six categories: (i) body temperature, (ii) blood oxygen saturation, (iii) heart rate monitoring, (iv) respiratory rate, (v) blood pressure, and (vi) simultaneous vital sign measurements. The Preferred Reporting Items for Systematic reviews and Meta-Analyses (PRISMA) methodology was used for the search and selection of scientific papers. The criteria to guide the scope of the review were defined with the Population, Intervention, Comparison, Outcomes, and Context (PICOC) methodology. The review highlighted significant efforts to develop and implement contactless, non-invasive devices for continuous monitoring outside clinical environments. It also revealed clear pathways for integrating AI at different stages of measurement and signal processing methods.