In fluid-structure interactions and fluid simulations, like incompressible two-phase flows involving high viscosity and density ratios, interface capturing and tracking is considered as a very challenging problem and has a strong impact on industrial applications. Usually, an adaptive grid is used to resolve the problem as well as to track the interface. In this paper, we describe an adaptive scheme based on the definition of an anisotropic metric tensor to control the generation of highly streched elements near an interface described with a level set function. In our approach, quasi-uniform anisotropic meshes are created with the objective of minimizing the interpolation errors by capturing the interface features using curvature-adapted anisotropic elements. The accuracy of the method is verified and numerical experiments are presented to show its efficiency.