An (even) 2-periodic cohomology theory or just periodic cohomology theory for short is an (even) multiplicative cohomology theory with a Bott element which is invertible (under multiplication in the cohomology ring of the point) so that multiplication by it induces an isomorphism
Compare with the notion of weakly periodic cohomology theory.
More generally one considers -periodic cohomology theories
The concept of even 2-periodic multiplicative cohomology theories originates with
The analogue of the Landweber exact functor theorem for even 2-periodic cohomology is discussed in