The author investigates regular semigroups by means of their bi-ideals. A bi-ideal of a semigroup S is a subsemigroup \(B\subseteq S\) with BSB\(\subseteq B\); the bi-ideals of S form a semigroup \({\mathfrak B}(S)\) under complex multiplication. The following two 'reduction theorems' are proven for \({\mathfrak B}(S):\) 1. If S is regular and \({\mathcal H}\) is a congruence on S then \({\mathfrak B}(S)\cong {\mathfrak B}(S/{\mathcal H})\). 2. If S is completely regular orthodox with band E of idempotents then \({\mathfrak B}(S)\cong {\mathfrak B}(E)\). Then various classes of completely regular orthodox semigroups are characterized by properties of their semigroups of bi-ideals. \{Reviewer's remark. Claim 1 above holds in the following sharper form: If \(\alpha\) is a congruence on a regular semigroup S and \(\alpha\subseteq {\mathcal H}\), then \({\mathfrak B}(S)\cong {\mathfrak B}(S/\alpha).\)\}