 In this report we shall study submanifolds in (pseudo-)Riemannian geometry and Finsler geometry. In Chapter 1, we use Gauss map to study the topology, volume and diameter of submanifolds in a sphere.We prove that if there exist 1 ≥ ε ＞ 0 and a fixed p-vector a such that the Gauss map g of an n-dimensional complete and connected submanifold M in Sn+p satisfies ＜ g, a ＞≥ ε,then M is diffeomorphic to Sn,and the volume and diameter of M satisfy εnvol(Sn) ≤vol(M) ≤ vol(Sn)/ε and επ ≤diam(M) ≤ π/ε,respectively. We also characterize the case where these inequalities become equalities.As an application,we obtain a differential sphere theorem for compact submanifolds in a sphere. In Chapter 2, we use Gauss map to study spacelike submanifolds in de Sitter space form. We prove that if there exist ρ ＞ 0 and a fixed unit simple (n + 1)-vector a ∈ Gpn+1,p such that the Gauss map g of an n-dimensional complete and connected spacelike submanifold Mn in Snp+p satisfies 〈g, a〉 ≤ ρ, then Mn is diffeomorphic to Sn,and its volume satisfies vol(Sn)/ρ ≤vol(M) ≤ ρnvol(Sn). We also characterize the case where these inequalities become equalities. In Chapter 3 we obtain a lower bound for the first Dirichlet eigenvalue of complete spacelike hypersurfaces in Lorentzian space in terms of mean curvature and the square length of the second fundamental form. This estimate is sharp for totally umbilical hyperbolic spaces in Lorentzian space. We also get a sufficient condition for spacelike hypersurface to have zero first eigenvalue in term of Gauss map. Then in Chapter 4, we deal with the Finsler geometry of submanifolds with respect to general Finsler volume element. The key is that Shens method still works in dealing with any other Finsler volume element, and we prove that there exists no closed oriented minimal submanifold in Minkowski space with respect to any Finsler volume element. We also obtain an estimate of volume growth for submanifolds in special Randers space and thus provides a necessary condition for a Randers space to be minimally immersed into special Randers space. Finally in Chapter 5, we obtain an extrinsic upper bound for the first eigenvalue of compact Finsler submanifolds in Minkowski space and obtain a Reilly type inequality. It is reduced to the standard Reilly inequality when the ambient space is Euclidean.