1. (1) Binding of single ligand at activity x to q binding sites on a non-dissociating protein can be characterized by q stepwise equilibrium constants ( K j ), a binding polynomial, N, and a saturation function y = (1/ q) d 1n N d 1n x. 2. (2) With two binding sites, 4 K 2 > K 1 is unambiguously positive cooperativity with Hill slope everywhere greater than unity. Also, N has complex conjugate roots, the semi-logarithmic plot is symmetrical and steeper than 1 4 at half saturation and concave up double reciprocal plots result. A stronger constraint (2 K 2 > K 1) is required for a sigmoid y( x) curve. 3. (3) Similarly, 4 K 2 < K 1 is unambiguously negative cooperativity with the binding polynomial giving real linear factors and the opposite curve shape features. A stronger constraint ( K 1 > 16 K 2) is necessary for a doubly sigmoid semi-logarithmic plot. 4. (4) For all higher degree cases, this classification is shown to break down and there is resulting confusion over the definition of cooperativity and the effect of this on curve shape. 5. (5) Complete analysis is possible in the third degree case and the conditions governing all possible definitions of cooperativity and the effects on factorability of the binding polynomial and curve shape are stated. 6. (6) When each successive binding is facilitated, the binding polynomial has complex conjugate roots and the Hill slope is greater than one. This is unambiguously positive cooperativity but can not be reliably diagnosed from the other graphical features found in the second degree case. 7. (7) When the binding of each successive ligand is inhibited, then the Hill slope is less than unity but the binding polynomial does not necessarily break up into real linear factors. This type of unambiguous negative cooperativity can not be reliably detected from the other specific graphical features. 8. (8) In all other cases involving mixed cooperativity there is ambiguity as to whether cooperativity should be defined in terms of successive cooperativity coefficients, factorability of the binding polynomial, semi-logarithmic or Hill slope or other curve shape features. 9. (9) In the third and all high degree cases it is always possible to decide upon the signs of 2 nK 2−( n−1) K 1 (initial cooperativity), 2 nK n −( n−1) K n−1 (final cooperativity) and n 2 K n − K 1 (overall operativity) from specific graphical features but the most theoretically useful plot for this and other classifications is the Hill plot. 10. (10) It is suggested that the most important feature for defining cooperativity is not the statistical ratios or the factorability of the binding polynomial as such but the sign and factorability of the Hessian of the binding polynomial.