A system \((f_i)_{i\in I}\) is a frame in a Hilbert space \(\mathcal H\) if there are positive constants \(A\) and \(B\) such that \(A\| f\| ^2 \leq \sum_{i\in I} |\langle f,f_i\rangle| ^2 \leq B\| f\| ^2\) for all \(f\in \mathcal H\). A system \(\mathcal V= ((V_i,V_i))_{i\in I}\) is a fusion frame or a frame of subspaces if \[ A\| f\| ^2 \leq \sum_{i\in I} v_i^2 \| \pi_{V_i}(f)\|^2 \leq B\| f\| ^2 \] where \(\pi_V\) is the orthogonal projection onto the subspace \(V\). One of the main results of this paper is a proof that the dual fusion frame \(((S_{\mathcal V}^{-1}V_i, v_i))_{i\in I}\) (with \(S_{\mathcal V}\) the frame operator given by \(\sum_{i\in I} v_i\pi_{V_i}(f)\)) is indeed a fusion frame. Other results deal with alternate duals, i.e., systems \(\mathcal W=((W_i,w_i))_{i\in I}\) so that \(f = \sum_{i\in I} v_iw_i \pi_{W_i}S_{\mathcal V}^{-1}\pi_{V_i}(f)\), and frame operators for a pair of two Bessel fusion sequences (where only the upper bound above is required to hold).