This paper studies termination properties of rewrite systems that are typeable using intersection types. It introduces a notion of partial type assignment on Curryfied Term Re\-write Systems, that consists of assigning intersection types to function symbols, and specifying the way in which types can be assigned to nodes and edges between nodes in the tree representation of terms. Two operations on types are specified that are used to define type assignment on terms and rewrite rules, and are proven to be sound on both terms and rewrite rules. Using a more liberal approach to recursion, a general scheme for recursive definitions is presented, that generalizes primitive recursion, but has full Turing-machine computational power. It will be proved that, for all systems that satisfy this scheme, every typeable term is strongly normalizable. Appeared as: @Inproceedings {Bakel-Fernandez-HOA'93, Author = "S. van Bakel and M. Fern\'andez", Title = "Strong {N}ormalization of {T}ypeable {R}ewrite {S}ystems", Booktitle = "Proceedings of HOA '93. First {I}nternational {W}orkshop on {H}igher {O}rder {A}lgebra, {L}ogic and {T}erm {R}ewriting, {\rm {A}msterdam, the {N}etherlands}. {S}elected {P}apers", Editor = "Jan Heering and Karl Meinke and Bernhard M{\"o}ller and Tobias Nipkow", Series = "Lecture Notes in Computer Science", Volume = "816", Publisher = "Springer-Verlag", Year = "1994", Pages = "20-39" }