This expository paper is concerned with the span of smooth closed manifolds (i.e. the maximum number of linearly independent vector fields) and with the corresponding notion of stable span. After a brief survey of older general results and methods the authors concentrate on (oriented) flag manifolds (which comprise e.g. (oriented) Grassmannians and, in particular, spheres and projective spaces) and on projective Stiefel manifolds. They give numerous upper and lower bounds and in some cases precise calculations for the span and the stable span of these specific manifolds.