It is well known that scalar curvature plays a fundamental role in general relativity. As its analogue, conformally variational Riemannian invariants (CVIs) is a category of fundamental scalar-type curvatures which shares many excellent properties in both conformal and Riemannian geometry. In this talk, I will give a brief introduction about our working frame about CVIs and present some interesting results about the geometric and analytic theory about CVIs. Hopefully, these theories can be helpful in the development of general relativity and other fields of mathematical physics. This talk is based on a series of work joint with Jeffrey S. Case from Penn State University and Yueh-Ju Lin from Wichita State University.