Recent advances in the studies of pure spin currents - flows of angular momentum (spin) not accompanied by the electric currents - have opened new horizons for the emerging technologies based on the electrons spin degree of freedom, such as spintronics and magnonics. The main advantage of pure spin current, as compared to the spin-polarized electric current, is the possibility to exert spin transfer torque on the magnetization in thin magnetic films without electrical current flow through the material. In addition to minimizing Joule heating and electromigration effects, this characteristic enables the implementation of spin torque devices based on the low-loss insulating magnetic materials, and offers an unprecedented geometric flexibility. Here we review the recent experimental achievements in investigations of magnetization oscillations excited by pure spin currents in different magnetic nanosystems based on metallic and insulating magnetic materials. We discuss the spectral properties of spin-current nano-oscillators, and relate them to the spatial characteristics of the excited dynamic magnetic modes determined by the spatially-resolved measurements. We also show that these systems support locking of the oscillations to external microwave signals, as well as their mutual synchronization, and can be used as efficient nanoscale sources of propagating spin waves.