The analysis of singular solutions plays an important role in many geometrical and physical problems, even if the problem one is interested in does not directly involve singular solutions,
as singular solutions may appear in the analysis of limits of regular solutions. In this talk, I will first survey a few earlier results involving the analysis of the asymptotic behavior of singular solutions to some conformally invariant equations, of which the Yamabe equation is a prototype. The analysis often has a global aspect and a local aspect, with the former involving the classification of entire solutions, or description of the singular sets, and the latter involving the local asymptotic behavior of the solution upon approaching the singular set. The two aspects are often closely related.  After the brief general survey, I will describe some recent results involving $\sigma_k$ curvature equations.