  We study reconstruction of an unknown function from its $d$-plane Radon
transform on the flat $n$-torus when $1 \leq d \leq n-1$. We prove new
reconstruction formulas and stability results with respect to weighted Bessel
potential norms. We solve the associated Tikhonov minimization problem on $H^s$
Sobolev spaces using the properties of the adjoint and normal operators. One of
the inversion formulas implies that a compactly supported distribution on the
plane with zero average is a weighted sum of its X-ray data.
