Power system is one of the largest man-made systems, which over the years has become the vital infrastructure for modern life. The foundation of today’s power system is three-phase alternating current (AC), which is adopted as the main stream technology for its advantages in voltage transforming, wide-area networking and long-distance power transmitting. The AC power system is a self-sustained oscillatory system operating at nominal frequency, i.e., 50 or 60 Hz. However, the multi-time-scale dynamics in a power system, mechanical, electrical or their coupled are prone to cause the exchange of energy at wide range of frequencies other than the nominal one, which is collectively known as the oscillation issues in a power system. Such oscillations would lead to power quality degradation, equipment failure, or even worse, system instability. Actually, the development history of power system is also a history of continuous “battle” against the oscillation problems. This article reviews the basic concepts and classification of the classical as well as emerging power system oscillations. The mechanism, origin and research advances of various oscillations are re-visited and discussed, from the earlier and long-standing low frequency oscillation (LFO), and then subsynchronous oscillation/resonance (SSO/SSR) associated with turbo-generators and wind turbine generators, to the latest wide-band oscillations involving the sophisticated dynamics of increasing renewables and power electronics in a power system. LFO incident was reported for the first time in 1964 in American Western Interconnection, where the formation of a larger power system by combining two smaller one, i.e., Northwest and Southwest systems, caused power oscillation at about 0.1 Hz. Later studies showed that the multiple machines with mechanical inertial but electrically connected through networks function like a mass-spring system, leading to LFOs at a frequency ranging from 0.01 to 2.5 Hz, depending on the size of the system. Initially, excitation control was improved by adding the function of power system stabilizer (PSS) to mitigate LFO. Later, advanced control, including linear optimal control, nonlinear control and optimal decentralized coordinated control, were developed to solve the problem. Since 2000, the continuous increase of system size and coverage causes very-low-frequency oscillation issues, to address which, several new technologies, for instance, wide-area monitoring and control, flexible AC transmission system (FACTS) control, has been commissioned in China and around the world. SSR/SSO is induced by the interactions between the multi-mass shaft of a turbo-generator and the series-compensated AC or HVDC transmissions. The frequency of power oscillation is subsynchronous, or less than the nominal frequency but generally above the 2.5 Hz. The first SSR incident was captured at Mohave Power generating Station in 1970. In China, however, SSR/SSO has been a severe stability concern since 2000, the time witnessing the widely usage of series compensation and HVDC to transfer electricity from remote coal-fueled power plants. After more than ten years’ continuous research, we have developed effective tools and countermeasures, for instance, multi-dimensional analysis, optimized control and coordinated protection, to solve practical SSR/SSO problems in China and some other countries. The integration of wind power introduces new interactive dynamics that involve wind turbine converters and AC grids. It is called sub-/super-synchronous control interaction (SSCI), which would result in a different type of oscillation from several Hz to double nominal frequency. Such sub/super-synchronous oscillations associated with wind power has entered people’s vision since 2009 and gained widespread attention after the occurrence of several actual incidents in US and China. To address this new issue, we have developed novel frequency-domain modeling and analysis methods to quantify the risk first, and then control strategies on either generator/converters side or grid side to modify the interaction to stabilize SSCI. These methods and strategies have been put into practice, for instance in Guyuan and Hami systems in China, to verify their effectiveness. At this moment, power system is undergoing profound changes, one of which is the rapid growth of renewables and power electronics in all sections, including generation, transmission and consumption of electricity. Such a change is triggering power system oscillations in a very wide band, i.e., from several Hz to several thousand Hz. The underlying reason is the multi-time-scale dynamic interactions among the huge number of converters and the increasingly complicated power grid. This leads to open challenges that are yet to be solved, including the need of: (1) Wide-spectrum models to represent the multi-time-scale dynamics; (2) advanced analysis methods to quantify the stability of wide-band oscillations; and (3) adaptive and coordinated controls to mitigate the oscillation under varying system conditions. In a nutshell, systematic and in-depth researches are required to establish new theories and methodologies for this emerging issue so as to promote the sustainable growth of renewable energy and the low-carbon development of power system.