The most famous of many problems in nonlinear analysis is Schauder's problem (Scottish Book, Problem 54) of the following form, that if $C$ is a nonempty convex compact subset of a linear topological space does every continuous mapping $f: C\to C$ has a fixed point? The answer we give in this paper is yes. In this connection this paper proves and extends the Markoff-Kakutani theorem to arbitrary linear topological space as an immediate consequence of the preceding solution of Schauder's problem. During the last twenty years this old conjecture was intensively examined by many mathematians. For set in normed spaces this has been proved by Schauder and for sets in locally convex spaces by Tychonoff. In this paper we prove that if $C$ is a nonempty convex compact subset of a linear topological space, then every continuous mapping $f: C\to C$ has a fixed point. On the other hand, in this sense, we extend and connected former results of Brouwer, Schauder, Tychonoff, Markoff, Kakutani, Darbo, Sadovskij, Browder, Krasnoselskij, Ky Fan, Reinermann, Hukuhara, Mazur, Hahn, Ryll-Nardzewski, Day, Riedrich, Jahn, Eisenack-Fenske, Idzik, Kirk, Göhde, Granas, Dugundji, Klee and some others.