Solar-C_EUVST (EUV High-Throughput Spectroscopic Telescope) is a solar physics mission concept that was selected as a candidate for JAXA competitive M-class missions in July 2018. The onboard science instrument, EUVST, is an EUV spectrometer with slit-jaw imaging system that will simultaneously observe the solar atmosphere from the photosphere/chromosphere up to the corona with seamless temperature coverage, high spatial resolution, and high throughput for the first time. The mission is designed to provide a conclusive answer to the most fundamental questions in solar physics: how fundamental processes lead to the formation of the solar atmosphere and the solar wind, and how the solar atmosphere becomes unstable, releasing the energy that drives solar flares and eruptions. The entire instrument structure and the primary mirror assembly with scanning and tip-tilt fine pointing capability for the EUVST are being developed in Japan, with spectrograph and slit-jaw imaging hardware and science contributions from US and European countries. The mission will be launched and installed in a sun-synchronous polar orbit by a JAXA Epsilon vehicle in 2025. ISAS/JAXA coordinates the conceptual study activities during the current mission definition phase in collaboration with NAOJ and other universities. The team is currently working towards the JAXA final down-selection expected at the end of 2019, with strong support from US and European colleagues. The paper provides an overall description of the mission concept, key technologies, and the latest status.