The diffraction-limited resolution of light focused by a lens was derived in 1873 by Est Abbe.Later in 1952,a method to reach sub-diffraction light spots was proposed by modulating the wavefront of the focused beam.In a related development,super-oscillating functions,that is,band-limited functions that lecally oscillate faster than their highest Fourier component,were introduced and experimentally applied for super-resolution microscopy.Up till now,only simple Gaussian-like sub-diffraction spots were used.Here we show that the amplitude and phase profile of these sub-diffraction spots can be arbitrarily controlled.In particular,we utilize Hermite-Gauss,Laguerre-Gauss and Airy functions to structure super-oscillating beams with subdiffraction lobes.These structured beams are then used for high-resolution trapping and manipulation of nanometer-sized particles.The trapping potential provides unprecedented localization accuracy and stiffness,significantly exceeding those provided by standard diffraction-limited beams.