  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.
