We investigate the geometry of codimension one foliations on smooth projective varieties defined over fields of positive characteristic with an eye toward applications to the structure of codimension one holomorphic foliations on projective manifolds. CONTENTS 1. Introduction 1 2. Distributions and foliations 2 Part 1. Geometry of foliations in positive characteristic 4 3. Differential geometry in positive characteristic 4 4. Particular features of foliations in positive characteristic 7 5. Degeneracy divisor of the p-curvature 13 6. Families of foliations 22 7. Integrability of the Cartier transform 24 8. Lifting the kernel of the p-curvature 26 Part 2. The space of holomorphic foliations on projective spaces 29 9. Foliations on complex projective spaces 29 10. Subdistributions of minimal degree 31 11. Codimension two subdistributions of small degree 34 12. Foliations without subdistributions of small degree 40 References 42