In this talk, we consider dimensional properties of the attractors of nonlinear and nonconformal iterated function systems (IFS) on R^d. We introduce a generalized transversality condition (GTC) for parameterized families of C^1 IFSs, and show that if the GTC is satisfied, then the Hausdorff and box-counting dimensions of the attractor of a typical IFS are given by the singularity dimension. Moreover, we verify the GTC for some parametrized families of C^1 IFSs. This is based on joint work with Karoly Simon.