The aim of this article is to present an overview of the current state of the mathematical and scientific competences in Primary Education in Spain and how their development is assessed in STEAM experiences (with students of 6–12 years of age). To this end, a systematic review based on the PRISMA 2020 protocol was carried out in the Scopus, Web of Sciences, and Dialnet databases, analyzing all articles published from 1993 to 2022 relating to the subject. In total, 53 articles were selected, most published in the last five years and primarily in the Dialnet database. It was found that the majority of the targeted objectives in the experiences were related to STEAM competence, but only 14 evaluated aspects related to mathematical or scientific competence. The mathematical competence was evaluated by the measurement of geometry or number content and their corresponding operations and by how information data (statistics) are graphically represented. In a few cases, the evaluation is also related to computational thinking and orientation skills. In the case of scientific competence, its evaluation was mainly based on the assessment of the understanding of concepts, with those concepts being related to Physics in 80% of the cases. The only evaluated aspect of scientific competence other than concept understanding was hypothesis formulation, which was featured in two articles. These results highlight the need to systematically incorporate assessments that provide a clearer view of mathematical or scientific competence development in implementing STEAM experiences in primary education.