Let (M,g) be a smooth, compact Riemannian manifold of dimension N\ge3 . We consider the almost critical problem (P_\epsilon)\qquad -\Delta_g u+ {N-2\over 4(N-1)}\mathrm {Scal}_g u= u^{{N+2\over N-2}+\epsilon }\quad\text{in}\ M\,,\quad u>0\quad\text{in}\ M\,, where \Delta_g denotes the Laplace-Beltrami operator, \mathrm {Scal}_g is the scalar curvature of g and \epsilon\in\mathbb R is a small parameter. It is known that problem (P_\epsilon) does not have any blowing-up solutions when \epsilon\nearrow0 , at least for N\leq 24 or in the locally conformally flat case, and this is not true anymore when \epsilon\searrow0 . Indeed, we prove that, if N\ge 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\left|\mathrm {Weyl}_g(\xi )\right|_g as \epsilon\searrow0 . Here Weyl _g denotes the Weyl curvature tensor of g.