Three viewpoints on the integral geometry of foliations

We deal with three different problems of the multidimensional integral geom etry of foliations. First, we establish asymptotic formulas for integrals o f powers of curvature of foliations obtained by intersecting a foliation by affine planes. Then we prove an integral formula for surfaces of contact o f an affine hyperplane with a foliation. Finally, we obtain a conformally i nvariant integral-geometric formula for a foliation in three-dimensional sp ace.