Let \(G\) be a reductive algebraic group defined over \(\mathbb{Q}\), suppose \(G\) acts on an affine space \(W\) over \(\mathbb{Q}\), and let \(V\) be an affine subvariety of \(W\) (defined over \(\mathbb{Q}\)) on which \(G\) acts transitively. The authors wish to find an asymptotic formula for \(N(T, V)= \text{card} (V(\mathbb{Z}) \cap B(T))\) as \(T\to \infty\), where \(B(T)= \{x\mid x\in W(\mathbb{R})\), \(|x|\leq T\}\) is the ball of radius \(T\) around the origin, \(|\cdot |\) being the usual Euclidean norm on \(\mathbb{R}^n\). If \(W(\mathbb{Z}) \cdot \Gamma \subseteq W(\mathbb{Z})\) for a subgroup \(\Gamma\) of finite index in \(G(\mathbb{Z})\) then \(V(\mathbb{Z})\) consists of a finite number of \(\Gamma\)-orbits, and the problem reduces to an evaluation of \(N(T, O)= \text{card} (O\cap B(T))\) for a \(\Gamma\)-orbit \(O= \Gamma\cdot y\), \(y\in V(\mathbb{Z})\). The authors introduce two conditions: (i) the quotient space \((Z(H) \cap \Gamma)\setminus Z(H)\) is compact, where \(H\) is the stabiliser of \(y\) and \(Z(H)\) is the centraliser of \(H\) in \(G(\mathbb{R})\), and a somewhat technical ``non-focussing'' condition (ii) on the set \(\{R (T) \mid T> 0\}\) of the open subsets \(R(T)= B(T) \cap V(\mathbb{R})\) of \(V(\mathbb{R})\). Assuming conditions (i) and (ii) are satisfied, they prove an asymptotic formula \(N(T, O)\sim \lambda (R(T))\) as \(T\to \infty\), where \(\lambda\) is a properly normalised \(G\)-invariant measure on \(V(\mathbb{R})\). Conditions (i) and (ii) are verified, in particular, for (a) \(V(\mathbb{Z})= \{X\mid X\in M_{n\times n} (\mathbb{Z})\), \(\text{det} (1- tX)= p(t)\}\), where \(p(t)\) is a fixed monic polynomial in \(\mathbb{Z} [t]\) of degree \(n\geq 2\), and (b) \(V(\mathbb{Z})= \{X \mid X\in M_{m\times n} (\mathbb{Z})\), \(XAX'= B\}\), where \(A\) is the matrix of an integral indefinite quadratic form of signature \((p, q)\), \(p+q= n\geq 3\), and \(B\) is the matrix of a positive definite integral quadratic form in \(m\) variables, \(m\leq \max \{p, q\}\) (here \(M_{m\times n} (\mathbb{Z})\) stands for the set of \(m\times n\) matrices with entries in \(\mathbb{Z}\)). In each of the cases (a) and (b), the authors obtain an explicit asymptotic formula for \(N(T, V)\). Furthermore, they construct a homogeneous space \(V\) which satisfies condition (i), but not (ii); in this case, the corresponding asymptotic formula differs from the one described above. These results are deduced from the authors' general theorem concerning convergence of probability measures on homogeneous spaces, whose proof uses the theory of unipotent flows on homogeneous spaces described in a series of recent papers by S. Dani, G. Margulis, M. Ratner, and N. Shah.