Small brightly fluorescent carbon nanoparticles have emerged as a new class of materials important for sensing and imaging applications. We analyze comparatively the properties of nanodiamonds, graphene and graphene oxide "dots", of modified carbon nanotubes and of diverse carbon nanoparticles known as "C-dots" obtained by different methods. The mechanisms of their light absorption and luminescence emission are still unresolved and the arguments are presented for their common origin. Regarding present and potential applications, we provide critical comparison with the other types of fluorescence reporters, such as organic dyes and semiconductor quantum dots. Their most prospective applications in sensing (based on the changes of intensity, FRET and lifetime) and in imaging technologies on the level of living cells and whole bodies are overviewed. The possibilities for design on their basis of multifunctional nanocomposites on a broader scale of theranostics are outlined.