This article analyses the motion of a classical relativistic string in flat complex ten-dimensional space-time. A general solution is presented for the equations of motion. The solution is given in terms of essentially freely specifiable functions. In deriving these results extensive use is made of spinors for ten dimensions, the basic properties of which are described in some detail. In particular, a significant role is played by those spinors in ten dimensions that satisfy the `purity' property of E. Cartan. These constitute an eleven-dimensional algebraic variety V in the sixteen-dimensional linear space of reduced (Weyl) spinors for the group SO(10). A general classical relativistic string in ten dimensions can be represented by means of a set of arbitrarily specifiable twice-differentiable curves in V. As a by-product of the investigation a general solution is also given for the equations of motion of a classical relativistic string in eight-dimensional space-time.