  We find a principle of harmonic analyticity underlying the quaternionic
(quaternion-K\"ahler) geometry and solve the differential constraints which
define this geometry. To this end the original $4n$-dimensional quaternionic
manifold is extended to a bi-harmonic space. The latter includes additional
harmonic coordinates associated with both the tangent local $Sp(1)$ group and
an extra rigid $SU(2)$ group rotating the complex structures. Then the
constraints can be rewritten as integrability conditions for the existence of
an analytic subspace in the bi-harmonic space and solved in terms of two
unconstrained potentials on the analytic subspace. Geometrically, the
potentials have the meaning of vielbeins associated with the harmonic
coordinates. We also establish a one-to-one correspondence between the
quaternionic spaces and off-shell $N=2$ supersymmetric sigma-models coupled to
$N=2$ supergravity. The general $N=2$ sigma-model Lagrangian when written in
the harmonic superspace is composed of the quaternionic potentials. Coordinates
of the analytic subspace are identified with superfields describing $N=2$
matter hypermultiplets and a compensating hypermultiplet of $N=2$ supergravity.
As an illustration we present the potentials for the symmetric quaternionic
spaces.
