  Given a complex algebraic variety X, we define a natural number called the
motivic dimension which measures the amount of transcendental (co)homology of
X. It is zero precisely when all the (co)homolgy is spanned by algebraic
cycles. Most of this paper is concerned with giving estimates on this number,
along with examples where it is small. As an application, we check or recheck
the Hodge conjectue in a number of examples: uniruled fourfolds, rationally
connected fivefolds, fourfolds fibred by surfaces with p_g=0, Hilbert schemes
of a small number points on surfaces with p_g=0, and generic hypersurfaces.
