Nucleation of new peptide and protein aggregates on the surfaces of amyloid fibrils of the same peptide or protein has emerged in the past two decades as a major pathway for both the generation of molecular species responsible for cellular toxicity and for the autocatalytic proliferation of peptide and protein aggregates. A key question in current research is the molecular mechanism and driving forces governing such processes, known as secondary nucleation. In this context, the analogies with other self-assembling systems for which monomer-dependent secondary nucleation has been studied for more than a century provide a valuable source of inspiration. Here, we present a short overview of this background and then review recent results regarding secondary nucleation of amyloid-forming peptides and proteins, focusing in particular on the amyloid β peptide (Aβ) from Alzheimer's disease, with some examples regarding α-synuclein from Parkinson's disease. Monomer-dependent secondary nucleation of Aβ was discovered using a combination of kinetic experiments, global analysis, seeding experiments and selective isotope-enrichment, which pinpoint the monomer as the origin of new aggregates in a fibril-catalyzed reaction. Insights into driving forces are gained from variations of solution conditions, temperature and peptide sequence. Selective inhibition of secondary nucleation is explored as an effective means to limit oligomer production and toxicity. We also review experiments aimed at finding interaction partners of oligomers generated by secondary nucleation in an ongoing aggregation process. At the end of this feature article we bring forward outstanding questions and testable mechanistic hypotheses regarding monomer-dependent secondary nucleation in amyloid formation.