The paper examines alternative postulates for subadditive processes, especially the ergodic theory thereof. It introduces superconvolutive sequences of distributions and proves limit laws for these, which generalize the weak law of large numbers, Chernoff's theorem, and Kesten's lemma. It discusses eigenshift and eigendistribution theory and concave recurrence relations in the convolutive semigroup, illustrating sundry conjectures with computer studies. It deals with applications of the theory to the first-death problem in branching processes, Bethe approximation of first-passage percolation, self-avoiding walks, maximal solutions of the generalized subconvolutive inequality, rates of convergence of a subadditive process, multidimensional subadditive processes in physics including the dimer problem and the overlapping-sphere model of liquid-vapor equilibrium, and Ulam's problem on the longest monotone subsequence of a random permutation.