  We give a complete classification of the infinite dimensional tilting modules
over a tame hereditary algebra R. We start our investigations by considering
tilting modules of the form T=R_U\oplus R_U /R where U is a union of tubes, and
R_U denotes the universal localization of R at U in the sense of Schofield and
Crawley-Boevey. Here R_U/R is a direct sum of the Pr\"ufer modules
corresponding to the tubes in U. Over the Kronecker algebra, large tilting
modules are of this form in all but one case, the exception being the Lukas
tilting module L whose tilting class Gen L consists of all modules without
indecomposable preprojective summands. Over an arbitrary tame hereditary
algebra, T can have finite dimensional summands, but the infinite dimensional
part of T is still built up from universal localizations, Pr\"ufer modules and
(localizations of) the Lukas tilting module. We also recover the classification
of the infinite dimensional cotilting R-modules due to Buan and Krause.
