Generalized homology theories, such as K-theory or bordism theory, have risen to great fame in topology for their usefulness as homotopical invariants as well as for their connections to more geometric concepts such as vector bundles and manifolds. Work of Quillen, Landweber, Morava, and others from the 1960s and 1970s has shown that there is an intimate relationship between certain classes of generalized homology theories and one-dimensional commutative formal groups. This has ushered in an era where much of homotopy theory is inspired from algebraic geometry, in particular elliptic curves, deformation theory, and modular or automorphic forms. While formal groups can thus be used to classify homology theories, I will go a step further and show how (unstable) operations in generalized cohomology theories can also be described as certain kinds of affine formal schemes with additional structure, namely as what I call formal plethories. Knowing a homology theory together with its operations is almost sufficient, at least in principle, to completely compute the unstable homotopy groups of spheres. I will show how the plethoric approach to operations plays out in this context and produces a fairly feasible computation of those homotopy groups for various classes of homology theories -- significantly simpler than what one would get if one used only ordinary, singular homology.
