Amyloids, fibrillar assembly of (poly)peptide chains, are associated with neurodegenerative illnesses such as Alzheimer's and Parkinson's diseases, for which there are no cures. The molecular mechanisms of the formation of toxic species are still elusive. Some peptides and proteins can form functional amyloid-like aggregates mainly in bacteria and fungi but also in humans. Little is known on the differences in self-assembly mechanisms of functional and pathogenic (poly)peptides. We review atomistic and coarse-grained simulation studies of amyloid peptides in their monomeric, oligomeric, and fibrillar states. Particular emphasis is given to the challenges one faces to characterize at atomic level of detail the conformational space of disordered (poly)peptides and their aggregation. We discuss the difficulties in comparing simulation results and experimental data, and we propose new simulation studies to shed light on the aggregation processes associated with amyloid diseases.