  In this paper we discuss the construction of non-geometric Q- and R-branes as
sources of non-geometric Q- and R-fluxes in string compactifications. The
non-geometric Q-branes, being obtained via T-duality from the NS 5-brane or
respectively from the KK-monopole, are still local solutions of the standard NS
action, where however the background fields G and B possess non-geometric
global monodromy properties. We show that using double field theory, redefined
background fields tilde G and beta as well as their corresponding effective
action, the Q-branes are locally and globally well behaved solutions.
Furthermore the R-brane solution can be at least formally constructed using
dual coordinates. We derive the associated non-geometric Q- and R-fluxes and
discuss that closed strings moving in the space transversal to the
world-volumes of the non-geometric branes see a non-commutative or a
non-associative geometry.
  In the second part of the paper we construct intersecting Q- and R-brane
configurations as completely supersymmetric solutions of type IIA/B
supergravity with certain SU(3) x SU(3) group structures. In the near horizon
limit the intersecting brane configurations lead to type II backgrounds of the
form AdS4 x M6, where the six-dimensional compact space M6 is a torus fibration
with various non-geometric Q- and R-fluxes in the compact directions. It
exhibits an interesting non-commutative and non-associate geometric structure.
Furthermore we also determine some of the effective four-dimensional
superpotentials originating from the non-geometric fluxes.
