An analogue R of the Dyer-Lashof algebra R and an analogue A of the Steenrod algebra A are defined for generalized homology and cohomology theories. It is shown that if there is an E_infinity-multiplicative structure on a spectrum H, then on the corresponding generalized cohomology H~*(X) of a topological space X there is an action A direct X H~* (X) -> H~* (X) of the Steenrod algebra, while if the space X is an E_infinity-space, then on the generalized homology H_* (X) there is an action R direct X H_* (X) -> H_* (X) of the Dyer-Lashof algebra. These actions are computed for cobordism of topological spaces. A connection is established between the Steenrod operations and the Landweber-Novikov operations.
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