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Spin Hall nano oscillator(SHNO),a new type spintronic nano-device,can electrically excite and control spin waves in both nanoscale magnetic metals and insulators with low damping by the spin current due to spin Hall effect and interfacial Rashba effect.Several spin-wave modes have been excited successfully and investigated substantially in SHNOs based on dozens of different ferromagnetic/nonmagnetic(FM/NM)bilayer systems(e.g.,FM = Py,[Co/Ni],Fe,CoFeB,Y3Fe5O12;NM = Pt,Ta,W).Here,we will review recent progress about spin-wave excitation and experimental parameters dependent dynamics in SHNOs.The nanogap SHNOs with in-plane magnetization exhibit a nonlinear self-localized bullet soliton localized at the center of the gap between the electrodes and a secondary high-frequency mode which coexists with the primary bullet mode at higher currents.While in the nanogap SHNOs with out of plane magnetization,besides both nonlinear bullet soliton and propagating spin-wave mode are achieved and controlled by varying the external magnetic field and current,the magnetic bubble skyrmion mode also can be excited at a low in-plane magnetic field.These spin-wave modes show thermal-induced mode hopping behavior at high temperature due to the coupling between the modes mediated by thermal magnon mediated scattering.Moreover,thanks to the perpendicular magnetic anisotropy induced effective field,the single coherent mode also can be achieved without applying an external magnetic field.The strong nonlinear effect of spin waves makes SHNOs easy to achieve synchronization with external microwave signals or mutual synchronization between multiple oscillators which improve the coherence and power of oscillation modes significantly.Spin waves in SHNOs with an external free magnetic layer have a wide range of applications from as a nanoscale signal source of low power consumption magnonic devices to spin-based neuromorphic computing systems in the field of artificial intelligence.