In this talk we introduce a novel method for solving the best multilinear rank approximation problem. Our algorithm differs from existing methods in two respects: (1) it exploits the fact that the problem may be viewed as an optimization problem over a product of Grassmann manifolds; (2) it uses Quasi-Newton-like Hessian-approximates specially adapted for Grassmannians and thus avoids the inevitable problem of large Hessians in such problems. Tensor approximation problems occur in various applications involving multidimensional data. The performance of the Quasi-Newton algorithm is compared with the Newton-Grassmann and Higher Order Orthogonal Iteration algorithms for general and symmetric 3-tensors.