Abstract The physical meaning and essence of Fresnel numbers are discussed, and two definitions of these numbers for off-axis optical systems are proposed. The universal Fresnel number is found to be $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}N=(a^{2}/\lambda z ) \ast C_{1} +C_{2} $ . The Rayleigh–Sommerfeld nonparaxial diffraction formula states that a simple analytical formula for the nonparaxial intensity distribution after a circular aperture can be obtained. Theoretical derivations and numerical calculations reveal that the first correction factor $C_{1} $ is equal to $\cos \theta $ and the second factor $C_{2} $ is a function of the incident wavefront and the shape of the diffractive aperture. Finally, some diffraction phenomena in off-axis optical systems are explained by the off-axis Fresnel number.