The book discusses new accelerated and vortex laser beams, which have been introduced into optics in the last 10 years. Accelerated beams are two-dimensional Airy beams. The main feature of the Airy beams is the curvature of the trajectory of the main maximum (main lobe). It propagates along a parabolic trajectory. Therefore, the Airy beams are called accelerating or ballistic (since a freely falling body moves along a parabola with constant acceleration). Other accelerated beams that propagate not along parabolic trajectories, but, for example, along a hyperbolic one, and others, are also considered. The method of transformation of slowing beams into accelerated ones is proposed. Three-dimensional Pearcey beams and Pearcey half-beams, which have the property of sharp (accelerated) self-focussing, are considered. Vortex laser beams were previously considered by the authors in the monograph ‘Vortex laser beams’ (Samara, Novaya teknika, 2012). In this book, new laser beams are considered that are not included in the monograph mentioned above. These are Hermite–Gauss vortex beams, asymmetric non-diffraction Bessel and Lommel modes, paraxial asymmetric Bessel–Gauss and asymmetric Laguerre–Gauss beams, and Hankel vector beams with linear and circular polarization. Analytical expressions for the orbital angular momentum are found for all vortex beams. All the considered vortex beams (except for the Hankel beams) depend on a continuous parameter, with which the orbital angular moment of the beam changes continuously. It can be whole or fractional. Many of the considered vortex laser beams were formed experimentally using a liquid-crystal light modulator. Vortex laser beams with an orbital angular momentum are used to increase the capacity of the information channels in wireless communication systems (the number of degrees of freedom in a laser beam increases), in quantum computer science to form a pair of photons entangled in the orbital 271angular momentum (the state of a photon with a fractional orbital moment is entangled), for sensing the atmosphere in the presence of turbulence (vortex beams are more resistant to distortion than the usual Gaussian beams) to capture and rotate micro-objects, including biological ones (vortex beams transmit their orbital angular momentum of a particle trapped and rotating it). This area of optics and photonics (laser beams, singular optics) is a hot topic booming, and this book is only an introduction to it.