Singularity theorems assuming trapped submanifolds of arbitrary dimension

New singularity theorems are proven in Lorentzian manifolds of arbitrary dimension n if they contain closed trapped submanifolds of arbitrary co-dimension. The timelike or null convergence conditions must be generalized to a condition on sectional curvatures, or tidal forces, which reduces to the former in the cases of co-dimension 1, 2 or n. Applications to higher dimensional theories and to the case of trapped circles are briefly mentioned.