Let (M,g) be a smooth, compact Riemannian manifold of dimension N \geq 3. We consider the almost critical problem (P_\epsilon) -\Delta_g u+ {N-2\over 4(N-1)} Scal_g u= u^{{N+2\over N-2}+\epsilon } in} M, u>0 in M, where \Delta_g denotes the Laplace-Beltrami operator, Scal_g is the scalar curvature of g and \epsilon \in R is a small parameter. It is known that problem (P_\epsilon) does not have any blowing-up solutions when \epsilon \to 0^-, at least for N \leq 24 or in the locally conformally flat case, and this is not true anymore when \epsilon \to 0^+. Indeed, we prove that, if N \geq 7 and the manifold is not locally conformally flat, then problem (P_\epsilon) does have a family of solutions which blow-up at a maximum point of the function \xi \to |Weyl_g(\xi)|_g as \epsilon \to 0^+. Here Weyl_g denotes the Weyl curvature tensor of g.