  We show that compatibility between the DAMA modulation result (as well as
less statistically significant excesses such as the CDMS Silicon effect and the
excess claimed by CRESST) with constraints from other experiments can be
achieved by extending the analysis of direct detection data beyond the standard
elastic scattering of a WIMP off nuclei with a spin--dependent or a
spin--independent cross section and with a velocity distribution as predicted
by the Isothermal Sphere model. To do so we discuss several new approaches for
the analysis of Dark Matter direct detection data, with the goal to remove or
reduce its dependence on specific theoretical assumptions, and to extend its
scope: the factorization approach of astrophysics uncertainties, the
classification and study of WIMP-nucleon interactions within non--relativistic
field theory, inelastic scattering and isovector-coupling cancellations
including subdominant two-nucleon NLO effects. Typically, combining two or more
of these ingredients can lead to conclusions which are very different to what
usually claimed in the literature. This shows that we are only starting now to
scratch the surface of the most general WIMP direct detection parameter space.
