Many nonlinear problems in physics, engineering, biology, and social sciences can be reduced to finding critical points. The first So far, we may say, to some extend, that there is an organized procedure for producing such critical points and these methods are called global variational and topological methods. Roughly speaking, the modern variational and topological methods consist of the following two parts. Minimax methods Ljusternik and Schnirelman in 1929 mark the beginning of global analysis, by which some earlier mathematicians no longer consider only the minima or maxima of variational integrals. In 1934, Ljusternik and schnirelman developed a method that seeks to get information concerning the number of critical points of a functional from topological data. These ideas are referred to as the Ljusternik-Schnirelman theory. One celebrated and important result in the last 30 years has been the mountain pass theorem due to Ambrosetti and Rabinowitz in 1973. Since then, a series of new theorems in the form of minimax have appeared via variouse linking, category and index theories. Now these results in fact become a wonderful tool in studying the existence of solutions to di_erential equations with variational Structures. Morse theory This approach towards a global theory of critical points was pursued by Morse in 1934. It reveals a deeprelation between the topology of spaces, the number and types of critical points of any function defined on it. This theory was highly successful in topology in the 1950s due to the e_orts of Milnor and Smale. In the works of Palais, male and Rothe, Morse theory was generalized to in_nite-dimensional spaces. By then it was recognized as a useful approach in dealing with di_erential equations and in particular, in _nding existence of multiple solutions. The critical group and Morse index also can be derived in some cases. Although there are some profound works on Morse theory and related topics, the applications are some what limited by the smoothness and nondegeneracy assumptions on the functionals. However, both minimax theory and Morse theory essentialy give answers on the existence of (multiple) critical points of a functional. They usually cannot provide many more additional properties of the critical point except some special pro_les such as the Morse index, critical groups, and so on. In this thesis we first characterize smooth functions on R with prescribed zero sets and prove that the closed subsets of R are all critical, but not all of them are properly critical. Then we provide a caractrization of properly critical subsets of the real line and use it to produce some properly critical subsets of higher dimensional Euclidean spaces. We particularly get the proper criticality of C n , where C n is the middle third Cantor set. Finally we characterize the critical sets of the sphere and of the closed cylinder.