Few studies have been performed on odd-order surfaces. Since odd-order surfaces consist of powers of absolute value of radial coordinates, they are rotationally invariant. Thus, one might infer that odd-order surfaces can be expressed as the form of ordinary aspherical surface (even-order power series). However, by considering the differentials of odd-order surfaces, Shibuya et al.^{5} proved that odd-order surfaces cannot be represented as an even-order power series. (This is due to the fact that the higher-order differentials of odd-order surfaces cannot be defined at the origin.) This property implies that odd-order surfaces have aberration characteristics different from those of ordinary even-order surfaces. Thus, they concluded that the odd-order surfaces are effective and practically confirmed that odd-order coefficients are effective parameters in optical design. However, this result seems to be contradictory with the completeness of Zernike polynomials because the rotational invariant terms of Zernike polynomials consist only of even-order monomials. Also, the fact that odd-order surfaces can be exactly represented by Zernike polynomials has not been proven. To resolve these problems, Tanabe et al.^{6} proved that not only the displacement but also slopes of odd-order surfaces are exactly represented by a finite number of Zernike polynomials. As a result, odd-order surfaces are exactly represented by a finite number of even-order polynomials. (Note that the impossibility of Taylor expansion, which means expansion of odd-order aspherical terms into even-order power series, is not contradictory with the completeness of Zernike polynomials.) They also practically confirmed that the effectiveness of this approximation for the case of Schmidt corrector plate with odd-order terms. However, since the required number of even-order terms, which closely approximates an odd-order surface, is not necessarily realistic for some cases, odd-order surfaces are effective in optical design. To analyze the characteristics of odd-order surfaces, we construct an aberration theory for odd-order surfaces in this paper. Moreover, we study how best to use them for optical design of extreme ultraviolet lithography (EUVL) cameras.