Alzheimer's disease is the most common of the protein misfolding ("amyloid") diseases. The deposits in the brains of afflicted patients contain as a major fraction an aggregated insoluble form of the so-called amyloid β-peptides (Aβ peptides): fragments of the amyloid precursor protein of 39-43 residues in length. This review focuses on biophysical studies of the Aβ peptides: that is, of the aggregation pathways and intermediates observed during aggregation, of the molecular structures observed along these pathways, and of the interactions of Aβ with Cu and Zn ions and with small molecules that modify the aggregation pathways. Particular emphasis is placed on studies based on high-resolution and solid-state NMR methods. Theoretical studies relating to the interactions are also included. An emerging picture is that of Aβ peptides in aqueous solution undergoing hydrophobic collapse together with identical partners. There then follows a relatively slow process leading to more ordered secondary and tertiary (quaternary) structures in the growing aggregates. These aggregates eventually assemble into elongated fibrils visible by electron microscopy. Small molecules or metal ions that interfere with the aggregation processes give rise to a variety of aggregation products that may be studied in vitro and considered in relation to observations in cell cultures or in vivo. Although the heterogeneous nature of the processes makes detailed structural studies difficult, knowledge and understanding of the underlying physical chemistry might provide a basis for future therapeutic strategies against the disease. A final part of the review deals with the interactions that may occur between the Aβ peptides and the prion protein, where the latter is involved in other protein misfolding diseases.