AbstractIn this article, the existence of additive BIB designs is discussed with direct and recursive constructions, together with investigation of a property of resolvability. Such designs can be used to construct infinite families of BIB designs. In particular, we obtain a series of B(sn,tsm, λt(tsm − 1) (sn‐m − 1)/[2(sm − 1)]) for any positive integer λ, such thatsn(sn − 1) λ ≡ 0 (modsm(sm − 1) and for any positive integertwith 2 ≤ t ≤ sn‐m, wheresis an odd prime power. Connections between additive BIB designs and other combinatorial objects such as multiply nested designs and perpendicular arrays are discussed. A construction of resolvable BIB designs withv = 4kis also proposed. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 235–254, 2007