Conical singularities occur quite often and naturally. For example, according to Cheeger-Colding, under Ricci curvature lower bounds, the limit spaces will generally carry singularity of conical type. This process of a family of smooth metrics limiting to a singular metric of conical type will be called conical degeneration. Again by Cheeger-Colding, under rather general conditions, the basic analytic quantities such as the eigenvalues and eigenfunctions will converge. So will the heat kernels (Ding). It is rather different story for global geometric invariants defined in terms of the eigenvalues. We will discuss some recent work in this direction.