The formation of aggregates from a range of normally soluble peptides and proteins is the hallmark of several neurodegenerative disorders, including Parkinson's and Alzheimer's diseases. Certain such aggregates possess the ability to replicate and spread pathology, within tissues and in some case also between organisms. An understanding of which processes govern the overall rate of aggregate formation is thus of key interest. Here, we discuss the fundamental molecular processes of protein aggregation, review how their rates can be determined by kinetic measurements in the test-tube, and explore the mechanistic similarities and differences to animal models and human disease. We conclude that a quantitative mathematical model for aggregate replication and spreading in vivo requires additional information but would provide a theoretical framework to understand results from different experiments and how they connect to human disease.