ANNALES POLONICI MATHEMATICI 80 (2003) Extension of separately holomorphic functions—a survey 1899–2001 by Peter Pflug (Oldenburg) Dedicated to Professor J´ ozef Siciak in honour of his 70th birthday Abstract. This note is an attempt to describe a part of the historical development of the research on separately holomorphic functions. 1. Introduction. First, let us fix some notations we will need in this survey article. Let N ∈ N, and let A j ⊂ D j ⊂ C k j , D j a domain, j = 1,...,N . The set X := X(A 1 ,...,A N ; D 1 ,...,D N ) := N  j =1 A 1 ×...×A j −1 ×D j ×A j +1 ×...×A N is called the N -fold cross associated to the N pairs (A j ,D j ). Observe that different pairs may lead to the same cross set; e.g. if N − 1 of the A j ’s coincide with the corresponding D j ’s, then X = D 1 × ... × D N . Moreover, let M ⊂ X (M = ∅ is allowed). For (a 1 ,...,a N ) ∈ A 1 × ... ×A N and 1 ≤ j ≤ N , we define the fiber of M over (a 1 ,...,  a j ,...,a N )( 1 ) as M (a 1 ,..., a j ,...,a N ) := {z j ∈ D j :(a 1 ,...,z j ,...,a N ) ∈ M }. We will always assume that all the fibers M (a 1 ,..., a j ,...,a N ) are closed in D j . Our aim is to study separately holomorphic functions. Recall that a function f : X(A 1 × ... × A N ; D 1 × ... × D N ) \ M → C 2000 Mathematics Subject Classification : 32D15, 32D10. Key words and phrases : separately holomorphic function. Research partially supported by the Nieders¨ achsisches Ministerium f¨ ur Wissenschaft und Kultur, Az. 15.3–50 113(55) PL. This is an extended version of the talk delivered at the Conference on Complex Anal- ysis, Bielsko-Bia la, 2001. ( 1 )(a 1 ,...,  a j ,...a N ) := (a 1 ,...,a j−1 ,a j+1 ,...,a N ). [21]