It is theoretically shown that by superimposing diffraction-free solutions of the Helmholtz equation, one can construct localized diffraction-free beams that pass through predetermined points on subwavelength distances. These beams are based on the phenomenon of superoscillations and thus do not contain any evanescent waves. The effect of an aperture and noise is examined in specific examples where truncated beams with lambda/3 subwavelength features can propagate into the far field. (C) 2011 Optical Society of America