Introduction. 1. Non-Negative Operators. 2. Differential Operators. 3. The Balakrishnan Operator. 4. An Extension of the Hirsch Functional Calculus. 5. Fractional Powers of Operators. 6. Domains, Uniqueness and the Cauchy Problem. 7. Negative and Imaginary Powers. 8. The Dore-Vennis Theorem. 9. Functional Calculus for Co-groups. 10. Imaginary Powers on Hilbert Spaces. 11. Fractional Powers and Interpolation Spaces. 12. Fractional Powers of some Differential Operators. Appendix. Notations. Bibliography.