The talk will consist of an exposition of certain recent results on the asymptotic dynamical behaviour of orbits of unipotent flows (actions of unipotent one-parameter subgroups) on homogeneous spaces of Lie groups and their application to certain problems in diophantine approximation, concerning especially values of quadratic forms. One of the main themes will be to consider, given a Lie group $G$ and a lattice $\Gamma$ in $G$, limits of sequences of normalised linear measures on segments of the form $\{u_tx|0\leq t\leq T\}$, where $\{u_t\}){t\in{\bf R}}$ is any unipotent one-parameter subgroup of $G$, $x\in G/\Gamma$ and $T>0$. The results are applied, in particular, to get asymptotic lower estimates for the number of integral solutions in bounded regions, for certain inequalities defined by quadratic forms. The underlying ergodic theory, the role of classification of invariant measures, the basic aspects of analysing closures and distributions of orbits of the flows in the ambient space and the approach to the number-theoretic problems will be discussed.