The following preprints are available from the WWW: Jaap van Oosten Realizability: A Historical Essay http://www.math.uu.nl/publications/preprints/1131.ps.gz Abstract: historical survey on Realizability. Focuses on notions of realizability used in the study of metamathematics of arithmetical theories, and topos-theoretic developments. Bibliography contains 96 items.24 pages Lars Birkedal and Jaap van Oosten Relative and Modified Relative Realizability http://www.math.uu.nl/publications/preprints/1146.ps.gz Abstract: We approach `relative realizability' from an abstract point of view, studying internal partial combinatory algebras in an arbitrary topos E. Let RT(E,A) denote the standard realizability topos over E w.r.t. A. We define the notion of `elementary subobject' in a topos; if, for two internal pca's A and B in E, there is an embedding which maps A as elementary subobject into B, there is a local geometric morphism from RT(E,B) to RT(E,A). Next we study the situation where an internal topology j is given; we have a tripos over E using only the j-closed subobjects of A, giving a topos RTj(E,A). RTj(E,A) is a subtopos of RT(E,A) and we have a pullback diagram of toposes: Sh_j(E)--->RTj(E,A) | | | | V V E ----> RT(E,A) If A--> B is an embedding with A elementary subobject of B, the local geometric morphism restricts to a local geometric morphism: RTj(E,B)-->RTj(E,A) Moreover if A --> B is a j-dense embedding, there is a logical functor (a filter-quotient situation): RTj(E,A) --> RTj(E,B). If j is an open topology, the inclusion RTj(E,A)-->RT(E,A) is open too, and it makes sense to consider its closed complement; we call this the modified relative realizability topos Mj(E,A) w.r.t. A and j. We have an automatic pullback diagram Sh_k(E)---->Mj(E,A) | | | | V V E ----> RT(E,A) where k denotes the closed complement of j. In a section `Examples' we show that our treatment generalizes former definitions of relative realizability (in Awodey, Birkedal, Scott) and modified realizability. 16 pages.