The generalized van der Waerden number \(w(k_0, \dots, k_{r-1};r)\) is the least positive integer \(n\) such that whenever \(\{1,2, \dots , n \}\) is partioned into \(r\) sets \(S_0,\ldots, S_{r-1}\) there is some \(i\) so that \(S_i\) contains a \(k_i\)-term progression. The authors determine several new exact values and list the number of different colourings in the extremal cases. They also give a general formula for \(w(k,2,2, \dots, 2;r)\) for sufficiently large \(k\).