This volume concerns invariants of G-torsors with values in mod p Galois cohomology - in the sense of Serre's lectures in the book Cohomological invariants in Galois cohomology - for various simple algebraic groups G and primes p. The author determines the invariants for the exceptional groups F4 mod 3, simply connected E6 mod 3, E7 mod 3, and E8 mod 5. He also determines the invariants of Spinn mod 2 for n = 12 and constructs some invariants of Spin14. Along the way, the author proves that certain maps in nonabelian cohomology are surjective. These surjectivities give as corollaries Pfister's results on 10- and 12-dimensional quadratic forms and Rost's theorem on 14-dimensional quadratic forms. This material on quadratic forms and invariants of Spinn is based on unpublished work of Markus Rost. An appendix by Detlev Hoffmann proves a generalization of the Common Slot Theorem for 2-Pfister quadratic forms.