The analyticity preservation principle is employed to demonstrate an impressive affinity between the field theories with intrinsic analytic structure (Yang and self-dual Yang-Mills theories, Kähler and hyper-Kähler gravities) and superfield gauge theories ( N = 1 and N = 2 Yang-Mills, N = 1 and N = 2 supergravities). The defining constraints of the former theories are interpreted as the integrability conditions for the existence of appropriate analytic subspaces and are solved by passing to the basis with manifest analyticity. We prefer to work within the analytic basis. This allows one, e.g., to replace the nonlinear splitting problem of the twistor approach by solving a linear equation. We begin with implications of familiar complex analyticity in Yang theory and Kähler gravity. A new development is the geometric interpretation of Kähler potential in an extended space with the central charge coordinate. Next, the analyticity of a different kind is introduced, the harmonic analyticity. It governs the geometry of self-dual Yang-Mills fields, in R 4 n ( n = 1, 2, …), specifying it in terms of an unconstrained prepotential which lives in the analytic harmonic space involving the sphere S 2. The harmonic analyticity determines also hyper-Kähler geometry (an accompanying paper) and has deep parallels in the twistor approach. Detailed comparison with the latter is made.