As of today, 5G is rolling out across the world, but academia and industry have shifted their attention to the sixth generation (6G) cellular technology for a full-digitalized, intelligent society in 2030 and beyond. 6G demands far more bandwidth to support extreme performance, exacerbating the problem of spectrum shortage in mobile communications. In this context, this paper proposes a novel concept coined Full-Spectrum Wireless Communications (FSWC). It makes use of all communication-feasible spectral resources over the whole electromagnetic (EW) spectrum, from microwave, millimeter wave, terahertz (THz), infrared light, visible light, to ultraviolet light. FSWC not only provides sufficient bandwidth but also enables new paradigms taking advantage of peculiarities on different EW bands. This paper will define FSWC, justify its necessity for 6G, and then discuss the opportunities and challenges of exploiting THz and optical bands.