In the present chapter we solve some boundary value problems of physical interest in the asymmetric environment that is described by the ellipsoidal geometry. More precisely, we solve: (i) the problem of thermal equilibrium of an ellipsoidal body, which was the problem that gave rise to the theory of ellipsoidal harmonics [223] as well as to the introduction of the general curvilinear system [228]; (ii) the problem of gravitational attraction by a homogeneous ellipsoid, which was an important problem of Newtonian Mechanics for many years and was finally solved by Jacobi, Gauss, Rodrigues, and others in the early nineteenth century [51]; (iii) the problem of an ellipsoidal perfect conductor [329]; (iv) the problem of the polarization potential, in terms of which the polarization tensor and the electric polarizability tensor are expressed [200, 286]; (v) the problem of the virtual mass potential in terms of which the virtual mass tensor and the magnetic polarizability tensor are expressed [200, 286]; and (vi) the problem of the generalized polarization potentials, in terms of which the general polarizability tensor is defined [216]. We also include a short section on the reduction of these solutions to the case of prolate and oblate spheroids, their asymptotic forms, and the sphere. General results on polarization tensors can be found in [5–10] as well as in [11]. Further references on boundary value problems in ellipsoidal geometry are [34, 35, 38, 56, 69, 82, 87, 120, 122, 139-142, 144, 145, 154-156, 185, 196, 205, 207, 218, 221, 222, 242, 247, 251-253, 261, 275, 277, 283, 289-293, 295, 297, 302, 304, 324, 325, 336, 362].