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We compute the quantum cohomology rings of the partial flag manifoldsF_{n_1\cdots n_k}=U(n)/(U(n_1)\times \cdots \times U(n_k)). The inductivecomputation uses the idea of Givental and Kim. Also we define a notion of thevertical quantum cohomology ring of the algebraic bundle. For the flag bundleF_{n_1\cdots n_k}(E) associated with the vector bundle E this ring is found.Comment: 33 page

Quantum cohomology of partial flag manifolds $$F_{n_1 } \ldots _{n_k }$$