Nanomagnets form the building blocks for a gamut of miniaturized energyefficient devices including data storage, memory, wave-based computing, sensors and biomedical devices. They also offer a span of exotic phenomena and stern challenges. The rapid advancements of nanofabrication, characterization and numerical simulations during last two decades have made it possible to explore a plethora of science and technology related to nanomagnet dynamics. The progress in the magnetization dynamics of single nanomagnets and one-and two-dimensional arrays of nanostructures in the form of dots, antidots, nanoparticles, binary and bicomponent structures and patterned multilayers have been presented in details.Progress in unconventional and new structures like artificial spin ice and three-dimensional nanomagnets and spin textures like domain walls, vortex and skyrmions have been presented. Furthermore, a huge variety of new topics in the magnetization dynamics of magnetic nanostructures are rapidly emerging. An overview of the steadily evolving topics like spatiotemporal imaging of fast dynamics of nanostructures, dynamics of spin textures, artificial spin ice have been discussed. In addition, dynamics of contemporary and newly transpired magnetic architectures such as nanomagnet arrays with complex basis and symmetry, magnonic quasicrystals, fractals, defect structures, novel three-dimensional structures have been introduced. Effects of various spin-orbit coupling and ensuing spin textures as well as quantum hybrid systems comprising of magnon-photon, magnon-phonon and magnon-magnon coupling, antiferromagnetic nanostructures are rapidly growing and are expected to dominate this research field in the coming years. Finally, associated topics like nutation dynamics and nanomagnet antenna are briefly discussed. Despite showing a great progress, only a small fraction of nanomagnetism and its ancillary topics have been explored so far and huge efforts are envisaged in this evergrowing research area in the generations to come.