Metrics and mechanisms: Measuring the unmeasurable in the science of science

•We introduce, compare, and discuss recently proposed contextualized and theory-derived science metrics.•We demonstrate how emerging metrics complement existing ones in accessing the performance of scientists and teams, expose hidden structures and mechanisms of science, and provide additional theoretical insights illuminating the foundations of the science of science.•We group metrics and theories by three fundamental properties of science: hot and cold science, soft and hard science, fast and slow science to remind us that understanding and modeling the mechanisms of science condition effective development and application of metrics.