Understanding the emergence or loss of enzyme functions comprises several approaches, such as genetic, structural, and kinetic studies. Promiscuous enzyme activities have been proposed as starting points for the emergence of novel enzyme functions, for example, through genetic models such as neofunctionalization and subfunctionalization. In both cases, neutral evolution would fix gene redundancy, critical in relaxing functional constraints and allowing specific mutations to drive innovation. The evolution of enzyme activities has a structural basis, with genetic mutations modifying the active site architecture, conformational dynamics, or interaction networks, which leads to the creation, enhancement, or restriction of enzyme functions where epistatic interactions are crucial. These structural changes impact the described kinetic mechanisms like ground-state stabilization (affinity), transition-state stabilization (catalysis), or a combination of both. Case studies across diverse enzyme families illustrate these principles, emphasizing the interplay between genetic, structural, and kinetic approaches. Finally, we discuss the importance of understanding evolutionary mechanisms and their impact on protein engineering and drug design for biomedical and industrial applications. However, these studies highlight that further experimental evolutionary data collection is necessary to enable the training of advanced machine learning models for use in biotechnological applications.