On a regular semigroup S natural order relations have been defined by Nambooripad and by Lallement. Different characterisations and relationships between the Nambooripad order J, Lallement's order λ and a certain relation k are considered in Chapter I. It is shown that on a regular semigroup S the partial order J is left compatible if and only if S is locally R-unipotent. This condition in the case where S is orthodox is equivalent to saying that E(S) is a left seminormal band. It is also proved that λ is the least compatible partial order contained in J and that k = λ if and only if k is compatible and k if and only if J is compatible. A description of λ and J in the semigroups T(X) and PT(X) is presented. In Chapter II, it is proved that in an orthodox semigroup S the band of idempotents E(S) is left quasinormal if and only if there exists a local isomorphism from S onto an R-unipotent semigroup. It is shown that there exists a least R-unipotent congruence on any orthodox semigroup, generated by a certain left compatible equivalence R. This equivalence is a congruence if and only if E(S) is a right semiregular band. The last Chapter is particularly concerned with the description of R-unipotent congruences on a regular semigroup S by means of their kernels and traces. The lattice RC(S) of all R-unipotent congruences on a regular semigroup S is studied. A congruence≡ on the lattice RC(S) is considered and the greatest and the least element of each ≡-class are described.