The aggregation of peptides into amyloid fibrils plays a crucial role in various neurodegenerative diseases. While it has been generally recognized that fibril formation in vivo may be greatly assisted or accelerated by molecular surfaces, such as cell membranes, little is known about the mechanism of surface-mediated fibrillation. Here we study the role of adsorbed Alzheimer's amyloid-beta peptide (A beta 42) on surface-mediated fibrillation using polymer coatings of varying hydrophobicity as well a supported lipid bilayer membrane. Using single molecule fluorescent tracking and atomic force microscopy imaging, we show that weakly adsorbed peptides with two-dimensional diffusivity are critical precursors to fibril growth on surfaces. This growth mechanism is inhibited on the highly hydrophilic surface where the surface coverage of adsorbed peptides is negligible or on the highly hydrophobic surface where the diffusion constant of the majority of adsorbed peptides is too low. Physical properties that favor weakly adsorbed peptides with sufficient translational mobility can locally concentrate peptide molecules on the surface and promote inter-peptide interaction via two-dimensional confinement, leading to fibrillation at A beta peptide concentration many orders of magnitude below the critical concentration for fibrillation in the bulk solution.