The worst case number of comparisons needed for sorting or selecting in rounds is considered. The following results are obtained. (a) For every fixed \(k\geq 2\), \(\Omega (n^{1+1/k}(\log n)^{1/k})\) comparisons are required to sort n elements in k rounds. \((O(n^{1+1/k}\log n)\) are known to be sufficient.) This improves the previously known bounds by a factor of (log n)\({}^{1/k}\), which separates deterministic algorithms from randomized ones, as there are randomized algorithms whose expected number of comparisons is \(O(n^{1+1/k}).\) (b) For every fixed \(k\geq 2\), \(\Omega (n^{1+1/(2^ k-1)}(\log n)^{2/(2^ k-1)})\) comparisons are required to select the median from n elements in k rounds. \((O(n^{1+1/(2^ k-1)}(\log n)^{2-2/(2^ k- 1)})\) are known to be sufficient.) This improves the previously known bounds by a factor of (log n)\({}^{2/(2^ k-1)}\) and separates the problem of finding the median from that of finding the minimum, as \(O(n^{1+1/(2^ k-1)})\) comparisons suffice for finding the minimum. (c) We show that ``approximate sorting'' in one round requires asymptotically more than \(c\cdot n \log n\) comparisons, for every constant c, and can be done in O(n log n log log n) comparisons. This settles a problem raised by Rabin.