  A general form of the Borel-Cantelli Lemma and its connection with the proof
of Khintchine's Theorem on Diophantine approximation and the more general
Khintchine-Groshev theorem are discussed. The torus geometry in the planar case
allows a relatively direct proof of the planar Groshev theorem for the set of
$\psi$-approximable points in the plane. The construction and use of Haudsorff
measure and dimension are explained and the notion of ubiquity, which is
effective in estimating the lower bound of the Hausdorff dimension for quite
general lim sup sets, is described. An application is made to obtain the
Hausdorff dimension of the set of approximable points in the plane when
$\psi(q)=q^{-v}$, $v>0$, corresponding to the planar Jarnik-Besicovich theorem.
