This paper presents a computational homogenization scheme that is of particular interest for problems formulated in curvilinear coordinates. The main goal of this contribution is to generalize the computational homogenization scheme to a formulation of micro-macro transitions in curvilinear convective coordinates, where different physical spaces are considered at the homogenized macro-continuum and at the locally attached representative micro-structures. The deformation and the coordinate system of the micro-structure are assumed to be coupled with the local deformation and the local coordinate system at a corresponding point of the macro-continuum. For the consistent formulation of micro-macro transitions, the operations scale-up and scale-down are introduced, considering the rotated representation of tensor variables at the different physical reference frames of micro- and macro-structure. The second goal of this paper is to use objective strain measures like the Green-Lagrange strain tensor for the solution of boundary value problems on the micro- and macro-scale by providing the required transformations for the work-conjugate stress, strain and tangent tensors into variables admissible for the considered micro-macro transitions and satisfying the averaging theorem. Copyright © 2007 John Wiley & Sons, Ltd.