The volume discrepancy of a point set \(S= \{{\mathbf r}_1, \dots, {\mathbf r}_n\}\), \({\mathbf r}_j = (r_j^{(i)})^s_{i=1}\) in the \(s\)-dimensional unit cube \(I^s\) is given by \(\sup \lambda (P)- \lambda (P_S)\), where \(\lambda\) is the Lebesgue measure on \(I^s\), \(P\) runs over all intervals in \(I^s\) and \(P_S\) is given by \[ P_S= \{{\mathbf x} \in P: x^{(i)} \leq r_j^{(i)},\;i=1, \dots,s, \quad \text{for some} \quad j \leq n\text{ with } {\mathbf r}_j \in P\}. \] With respect to this and related notions discrepancy estimates are given. In particular the Van der Corput sequence is considered for dimension one and special sequences with small volume discrepancy are constructed in dimension two.