We present, in our knowledge, the first theoretical demonstration of the possibility to trap and manipulate particles in three dimensions with the radiation pressure exerted by a single acoustical beam. Numerical examples demonstrate that single-beam acoustical tweezers operating in three dimensions are feasible with a large variety of materials and may widely extend the range of forces and operation regions that are currently available with optical tweezers. To do so, a method to model the focusing properties of acoustical beams with complex wavefronts using a spherical transducer is proposed. Then, the radiation forces exerted by various beams going from the classical vortex to the high radial degree spherical vortex beam that we introduce here are studied. While the first is shown to trap moderately small particles, the latter will stiffly trap large solid spheres in three dimensions. Even though this demonstration is carried out using a formalism suited to acoustics, it is easily applicable to trap non-transparent particles with optical tweezers that remain an issue. (C) 2013 AIP Publishing LLC.