The cutoff wave numbers knm and the field of two-conductor, perfectly conducting wave guides are determined analytically. Three types of wave guides are considered: Eccentric circular conductors of radii R1, R2 and distance d between their axes, elliptic inner with circular outer conductor, and circular inner with elliptic outer conductor. The electromagnetic field is expressed in the first case in terms of circular cylindrical wave functions referred to both axes in combination with related addition theorems, while in the last two cases, both circular and elliptical cylindrical wave functions are used, which are further connected with one another by well-known expansion formulas. When the solutions are specialized to small eccentricities, kd in the first case and h = ka/2 in the last two cases (where a is the interfocal distance of the elliptic conductor), exact, closed-form expressions are obtained for the coefficients gnm in the resulting relations knm(d) = knm(0)[1 + gnm(knmd)2 + ···] and knm(h) = knm(0) [1 + gnmh2 + ···] for the cutoff wave numbers of the corresponding wave guides. Similar expressions are obtained for the field. Numerical results for all types of modes, comparisons, and certain generalizations are also included.