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Hochschild cohomology behaves well over a field, and its derived analogue Shukla cohomology behaves well over any base commutative ring. Both are intimately related to deformation theory. To study `nonlinear' deformations (e.g. Z/p^2 over Z/p), one wants to study Mac Lane cohomology, which introduces nonadditive features. Mac Lane cohomology ought to be the same thing as topological Hochschild cohomology; the analogue for homology is known by work of Pirashvili and Waldhausen. I'll give a quick recap on topological Hochschild cohomology, which is morally just Shukla cohomology with base `ring' the sphere spectrum. I'll then give a definition of THH^* for schemes, along with some comparison theorems showing that for reasonable schemes, any of the `obvious' definitions that one might make all agree. I'll give some (easy!) computations of THH^* for P^1 and P^2 over a finite field.