  The work develops further the theory of the following inversion problem,
which plays the central role in the rapidly developing area of thermoacoustic
tomography and has intimate connections with PDEs and integral geometry: {\it
Reconstruct a function $f$ supported in an $n$-dimensional ball $B$, if the
spherical means of $f$ are known over all geodesic spheres centered on the
boundary of $B$.} We propose a new unified approach based on the idea of
analytic continuation. This approach gives explicit inversion formulas not only
for the Euclidean space $\bbr^n$ (as in the original set-up) but also for
arbitrary constant curvature space $X$, including the $n$-dimensional sphere
and the hyperbolic space. The results are applied to inverse problems for a
large class of Euler-Poisson-Darboux equations in constant curvature spaces of
arbitrary dimension.
