Several properties of Hausdorff continua are considered in this paper. However, the major emphasis is on developing the properties of the hyperspaces 2x and C(X) of a Hausdorff continuum X. Preliminary definitions and notation are introduced in Chapter I. Chapters II and III deal with the topological structure of the hyperspaces and the concept of topological convergence. Properties of 2x and C(X) are investigated in Chapter IV, while Chapters V and VI are devoted to the Hausdorff continuum X. Chapter VII consists of theorems pertaining to Whitney maps and order arcs in 2x. Examples of C(X) are provided in Chapter VIII. Inverse sequences of Hausdorff continua and of their hyperspaces are considered in Chapter IX.