Publisher Summary This chapter discusses the curves and surfaces in the topologic spaces, with emphasis on the digital case. The Jordan curves are defined by parameterization. The Urysohn-Menger curves can be defined using a topologic approach. The two definitions are equivalent and both of them define separations of the plane. In addition, in picture analysis, one has to deal with curves that are given in digitized pictorial form and for which a parametric description is often not of interest. Topologic methods of defining curves are therefore more relevant for purposes. The chapter further discusses elementary curves and the Euler characteristics. An elementary curve is the union of a finite number of simple arcs, each pair of which have at most a finite number of points in common. It consists of a finite number of singular points and a finite number of regular components; the latter are either simple curves or simple arcs.