The theory of elasticity in two dimensions is generalized to include anisotropy. It is found that six coefficients are required to fully characterize the elastic response of anisotropic films. The theory is applied to monolayers with specific boundary conditions which represent cases of particular interest in the study of Langmuir-Blodgett films. In the special case vt hen an anisotropic film possesses two orthogonal symmetry pianos, only four constants are needed to fully characterize the elastic behavior of the film. Several experimental procedures are outlined to obtain the elastic constants experimentally.