Biconcavity is a simple condition on inverse demand that corresponds to the ordinary concept of concavity after simultaneous parameterized transformations of price and quantity. The notion is employed here in the framework of the homogeneous-good Cournot model with potentially heterogeneous firms. The analysis leads to unified conditions, respectively, for the existence of a pure-strategy equilibrium via nonincreasing best-response selections, for existence via quasiconcavity, and for uniqueness of the equilibrium. The usefulness of the generalizations is illustrated in cases where inverse demand is either "nearly linear" or isoelastic. It is also shown that commonly made assumptions regarding large outputs are often redundant.