the renormalization group
Stand back from a system and squint, blurring out the finest details, then ask what the world looks like at the new, coarser scale. Do this repeatedly and watch how the effective description evolves. This deceptively simple idea — systematically eliminating short-distance degrees of freedom and seeing how the theory changes with scale — is the renormalization group, and it is the tool that finally explained why critical exponents take the values they do and why universality exists at all.
An RG transformation coarse-grains the system (for example, Kadanoff's block-spin construction, which replaces each block of spins by a single effective spin), then rescales lengths back to the original spacing and renormalizes the fields. The net effect is a map on the space of coupling constants: each step sends one set of couplings to another, generating a flow through 'theory space'. Fixed points of this flow — where the couplings map to themselves — are scale-invariant, and a critical point sits exactly at such a fixed point. Linearizing the flow near a fixed point classifies perturbations as relevant (they grow under coarse-graining and drive the system away from criticality), irrelevant (they shrink, and their washing-out is the origin of universality), or marginal (borderline). The critical exponents are then read directly off the eigenvalues of the linearized RG transformation at the fixed point. Kenneth Wilson developed this into a calculational method (Nobel Prize 1982).
The renormalization group explains critical exponents and universality from first principles, and the same framework governs the 'running' of coupling constants in quantum field theory, where it explains asymptotic freedom in the strong interaction. An honest caveat about the name: the RG is really a SEMIGROUP, because coarse-graining discards information and cannot be inverted — you cannot run the flow backward to recover the fine-grained theory. 'Group' is a historical misnomer. And in practice the flow is computed only approximately, through schemes like the epsilon-expansion around four dimensions or Monte Carlo simulation.
In the block-spin picture of the 2D Ising model, one groups spins into blocks and assigns each block a single spin by majority rule, then rescales. Iterating this flow, a system exactly at T_c stays put at the critical fixed point, while any system slightly off T_c flows steadily away toward either the all-ordered or all-disordered fixed point.
Under RG flow the critical point is a fixed point; nearby systems flow away from it.
Despite the name, the RG is a semigroup: coarse-graining loses information and is not invertible. Critical exponents come out as eigenvalues of the linearized flow at a fixed point, and irrelevant couplings flowing to zero are precisely why microscopically different systems share the same exponents (universality).