The concept of Fresnel number is discussed and expressions are derived for misaligned optical systems. For the case of perfectly aligned optical elements, the usual number,N, based on a Fresnel zone concept is found to be given byN = (a 2/λ)(D 1/B 1 +A 2/B 2), whereB 1,D 1 andB 2,A 2 are the transfer matrix elements of the optical systems before and after a circular aperture of radiusa respectively. A modified definition of the Fresnel number is proposed for Gaussian beam propagation. This parameterN′ G, is related to the complex beam parameter and may be represented by the angleθ = tan−1 N′ G, found in the familiar Collins chart and its dual representation. A general relation for the transformation of this Fresnel number is found. The expressions for Gaussian beam transformation are thus simplified, since the waist-waist transform is given byN′ G1 =N′ G2 = 0. Finally, these two kinds of Fresnel numbers are written as tensors when applied to cases involving elliptical apertures, astigmatic beams and nonsymmetrical systems.