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