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The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1). It was finalized by Alexandre Grothendieck, during the 1950s and the 1960s.
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Schemes for conditional distributions (monads)
Schemes make it possible to think of varieties over different fields as manifestations of the same underlying object, even though they might appear very different and contain different amounts of information …