Let R be a Noetherian domain and let Ah(R) = {k E NI 3P, 0 E Spec R, P c Q, height 0 = height P + height Q/P+ k} the set of abnormalities of R ; notice that R is catenarian if and only if Ah(R) = (0). It is known that Ah(R) needs not be reduced to (0); indeed, Nagata constructed in [8] a local domain D that does not satisfy the altitude formula, so that R = D[X] is not catenarian [lo, Theorem 3.6, p. 5211. Houston-McAdam raised the question of the possibility for Ah(R) to be an infinite set [4, p. 671. We solve this problem in Section 3: for every set S such that (0) c S E N, we construct a Noetherian domain D such that Ab(D[X]) = S; this D has the additional property of being catenarian. Of course, when S is an infinite set, D must be infinite dimensional and consequently cannot be local. If D is an n-dimensional local domain, then D[X] is (n + l)-dimensional and Ab(D[Xl) E (0, 1, . . . , n 1). The problem here is whether or not for every set S such that{O}cSE{O,l,..., n l}, there exists an n-dimensional local domain D such that Ab(D[X])= S. We answer this problem affirmatively in Section 2; however, the constructed D is not in general catenarian; as a matter of fact, if for some n 5 3 and some (0)~s c (0, 1, . . . , n 1) there existed such a local catenarian then it would be a counterexample to several existing conjectures, as for instance the upper conjecture [5, p. 7501. In Section 3 we then show that for every set S such that {o}~sc{o,1,..., n l}, there exists a catenarian n-dimensional semi local domain D such that Ab(D[X]) = S. A related problem to the preceding one is whether or not for every n 2 2 and Ts{2,..., n} there exists a local domain D, with maxima1 idea1 M such that T = {r / there exists a saturated chain of prime ideals of length r between (0) and M}. We also answer this problem affirmatively in Section 2, providing a wide class of non catenarian local domains with prescribed lengths for chains of prime ideals. For the construction of the examples of Sections 2 and 3 we develop a “glueing process” which generalizes a device that appears in Grothendieck [3, pp. lOl-1021 and in Nagata [8]. The main advantage of this generalization over Nagata’s device is