Gamma-ray bursts (GRBs) are short-duration flashes of gamma rays occurring at cosmological distances and the most violent explosive phenomena since the cosmic big bang. GRBs were accidentally discovered by Vela military satellites of United States in 1967. The BATSE instrument onboard the Compton Space Gamma-Ray Observatory found two types of GRBs, long-duration ($T_{90}>$ 2s) and short-duration ($T_{90} Science as one of the top ten scientific breakthroughs of the year. In 1999, long GRBs were found to be associated with the birth of stellar mass black holes. In 2003, the HETE-II satellite discovered the first direct association of a long GRB with a type Ic supernova, confirming that long GRBs are linked to the core collapse of massive stars. Science ranked these two major advances as one of the worlds top ten scientific breakthroughs of the year in 1999 and 2003, respectively. In 2005, the Swift satellite made the first accurate localization of short GRBs, leading to the discoveries of afterglows, host galaxies and redshifts of short GRBs. These observations provide indirect evidence that short GRBs originate from mergers of binary systems of compact objects (at least including one neutron star) at cosmological distances. In particular, on 2017 August 17, the LIGO/Virgo gravitational wave (GW) detectors, for the first time, discovered a GW event from a binary neutron star (BNS) merger, GW170817. About 1.74 s after the merger, Fermi/GBM detected a short gamma-ray burst (named GRB170817A). Subsequently, many ground-based and space-based telescopes detected X-ray, ultraviolet, optical, nearly infrared, and radio counterparts to GW170817, especially including a multi-wavelength kilonova (named AT2017gfo). These discoveries mark the beginning of a new era of multi-messenger astronomy. To summarize, all the observations have shown that long bursts originate from the core collapse of massive stars and short bursts originate from the mergers of binary compact objects (at least including one neutron star); besides prompt gamma-ray emission, the sources of GRBs produce X-ray, optical and radio afterglows in timescales of weeks, months and years after the burst trigger, respectively. Theoretically, prompt gamma-ray emissions of GRBs are thought to arise from some energy dissipation processes in the interiors of relativistic jets and multi-wavelength afterglows arise from forward shocks due to collisions between the jets and their ambient media. Therefore, GRBs are not only astronomical laboratories of studying extremely physical phenomena (e.g., newborn compact objects including stellar-mass black holes and neutron stars, gravitational waves, ultra-high-energy cosmic rays, and high-energy neutrinos) and of testing the basic physical principles with high accuracy, but also become an important probe of the star formation and evolution in the early universe, high-redshift galaxies, and cosmology. GRBs now are a multidisciplinary field (including astronomy, cosmology, and physics) and thus one of the most competitive fundamental research fields. In this paper, we review recent researches of GRBs and electromagnetic counterparts to gravitational waves, by focusing on the relevant key scientific issues, and discuss how to seize the opportunity to plan the interdisciplinary strategy based on the development trend and the research foundation in China, to maximize domestic scientific equipment achievements, and to enhance Chinese international influence in this field.