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A Peterson variety is a subvariety of the flag variety $G/B$ which appears inthe construction of the quantum cohomology of partial flag varieties. EachPeterson variety has a one-dimensional torus $S^1$ acting on it. We give abasis of Peterson Schubert classes for $H_{S^1}^*(Pet)$ and identify the ringgenerators. In type $A$ Harada-Tymoczko gave a positive Monk formula, andBayegan-Harada gave Giambelli's formula for multiplication in the cohomologyring. This paper gives Monk's rule and Giambelli's formula for all Lie types.

Monk's Rule and Giambelli's Formula for Peterson Varieties of All Lie Types