Iwahori Hecke Algebras and their Representation Theory : Lectures given at the C.I.M.E. Summer School held in Martina Franca Italy June 28 July 6 1999 Preface I. Cherednik and Ya. Markov: Hankel transform via double Hecke algebra R. Howe: Affine like Hecke algebras and p adic representation theory G. Lusztig: Notes on affine Hecke algebras. Two basic problems of representation theory are to classify irreducible representations and decompose representations occuring naturally in some other context. Algebras of Iwahori Hecke type are one of the tools and were probably first considered in the context of representation theory of finite groups of Lie type. This volume consists of notes of the courses on Iwahori Hecke algebras and their representation theory given during the CIME summer school which took place in 1999 in Martina Franca Italy.