Table of Contents -- Introduction -- 1. Smooth projections -- 2. Projective units and projective faces -- 3. Geometric properties of projective faces -- 4. The lattice of P-projections -- 5. Compatibility -- 6. Existence of spectral resolutions -- 7. Uniqueness of spectral resolutions -- 8. Functional calculus -- 9. Abelian subspaces -- 10. Spectral convex sets -- 11. Applications to operator algebras -- 12. Some further results on spaces in spectral duality and on Jordan structure -- References