  We compute the matching conditions for a general thick codimension 2 brane, a
necessary previous step towards the investigation of gravitational phenomena in
codimension 2 braneworlds. We show that, provided the brane is weakly curved,
they are specified by the integral in the extra dimensions of the brane
energy-momentum, independently of its detailed internal structure. These
general matching conditions can then be used as boundary conditions for the
bulk solution. By evaluating Einstein equations at the brane boundary we are
able to write an evolution equation for the induced metric on the brane
depending only on physical brane parameters and the bulk energy-momentum
tensor. We particularise to a cosmological metric and show that a realistic
cosmology can be obtained in the simplest case of having just a non-zero
cosmological constant in the bulk. We point out several parallelisms between
this case and the codimension 1 brane worlds in an AdS space.
