In this article, we treat stability conditions in the sense of King, Bridgeland and Bayer in a single framework. Following King, we begin with weight functions on a triangulated category, and consider increasingly specialised configurations of triangulated categories, t-structures and stability functions that give equivalent categories of stable objects. Along the way, we recover existing results in representation theory and algebraic geometry, and prove a series of new results on elliptic surfaces, including correspondence theorems for Bridgeland stability conditions and polynomial stability conditions, local finiteness and boundedness for mini-walls for Bridgeland stability conditions, isomorphisms between moduli of 1-dimensional twisted Gieseker semistable sheaves and 2-dimensional Bridgeland semistable objects, the preservation of geometric Bridgeland stability by autoequivalences on elliptic surfaces of nonzero Kodaira dimension, and solutions to Gepner equations on elliptic surfaces.
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