  The non-perturbative quantum geometry of the Universal Hypermultiplet (UH) is
investigated in N=2 supergravity. The UH low-energy effective action is given
by the four-dimensional quaternionic non-linear sigma-model having an U(1)xU(1)
isometry. The UH metric is governed by the single real pre-potential that is an
eigenfunction of the Laplacian in the hyperbolic plane. We calculate the
classical pre-potential corresponding to the standard (Ferrara-Sabharwal)
metric of the UH arising in the Calabi-Yau compactification of type-II
superstrings. The non-perturbative quaternionic metric, describing the
D-instanton contributions to the UH geometry, is found by requiring the SL(2,Z)
modular invariance of the UH pre-potential. The pre-potential found is unique,
while it coincides with the D-instanton function of Green and Gutperle, given
by the order-3/2 Eisenstein series. As a by-product, we prove cluster
decomposition of D-instantons in curved spacetime. The non-perturbative UH
pre-potential interpolates between the perturbative (large CY volume) region
and the superconformal (Landau-Ginzburg) region in the UH moduli space. We also
calculate a non-perturbative scalar potential in the hyper-K"ahler limit, when
an abelian isometry of the UH metric is gauged in the presence of D-instantons.
