The propagation of solitons along the interface between two dielectric nonlinear media was investigated theoretically extensively in the 1980s but never realized experimentally. Recently we predicted that the required small index differences between the media and hence solitons can be created at the interface between continuous and periodic discrete media consisting of arrays of weakly coupled waveguides. Our theoretical analysis has predicted the existence of stable solitons with power thresholds both in the centre and at the edge of the Brillouin zone. We have observed both of these discrete surface solitons with power thresholds in both Kerr and quadratically nonlinear media. Spatial solitons with fields in neighboring channels either in phase or pi out of phase with one another have been identified.