We present a set of optimal and asymptotically optimal sequential and parallel algorithms for the problem of searching on an m × n sorted matrix in the general case when m⩽n. Our two sequential algorithms have a time complexity of 0(mlog(2nm)) which is shown to be optimal. Our parallel algorithm runs in 0(log(logmlog log m) log (2nm1-z)) time using m/log(logmlog logm) processors on a COMMON CRCW PRAM, where 0 ⩽ z < 1 is a monotonically decreasing function on m, which is asymptotically work-optimal. The two sequential algorithms differ mainly in the ways of matrix partitioning: one uses row-searching and the other applies diagonal-searching. The parallel algorithm is based on some non-trivial matrix partitioning and processor allocation schemes. All the proposed algorithms can be easily generalized for searching on a set of sorted matrices.