universality
Boiling water and a magnet losing its magnetism have nothing microscopically in common — one is molecules escaping a liquid, the other is atomic spins randomizing. Yet near their critical points they behave in quantitatively identical ways, described by the very same power laws with the very same exponents. This astonishing indifference to microscopic detail is called universality, and it is one of the deepest and most beautiful discoveries in statistical physics.
Universality is the statement that the critical behavior of a system — its critical exponents and its scaling functions — depends only on a few gross features: the spatial dimension d, the number of components and symmetry of the order parameter, and the range of the interactions. It does NOT depend on the microscopic details: the lattice structure, the precise form of the short-range forces, the chemical identity of the particles. Systems sharing these gross features fall into the same universality class and have identical critical exponents. The canonical example is the 3D Ising class, which contains the uniaxial ferromagnet, the liquid-gas critical point, and the demixing of certain binary alloys and fluid mixtures — all with beta about 0.33 and gamma about 1.24.
Universality is explained by the renormalization group: as one coarse-grains a system, the microscopic couplings that distinguish one member of a class from another turn out to be 'irrelevant' and flow away, while only a few 'relevant' couplings (essentially temperature and field) survive to control the critical point. This is why the whole class flows to a common fixed point with common exponents. An honest limitation: universality governs only the leading singular behavior right at criticality; the value of T_c, the non-universal amplitudes, and the far-from-critical behavior all depend on microscopic specifics.
Change the shape of the lattice from square to triangular, or swap the atoms in a magnet, and T_c shifts — but the critical exponents do not budge, as long as the dimensionality and order-parameter symmetry are unchanged. That invariance is universality: the microscopic details wash out, leaving only d and the symmetry to matter.
Only dimensionality and symmetry survive; microscopic details are washed out at criticality.
Universality applies to critical exponents and scaling functions, NOT to non-universal quantities like T_c or amplitudes, which do depend on microscopic details. What defines a universality class is dimensionality plus the symmetry of the order parameter, not the specific material.