Short answers, pulled from the story.
General equilibrium theory attempts to explain supply, demand, and prices across a whole economy with many interacting markets, and to show when their interaction produces an overall equilibrium. It contrasts with partial equilibrium theory, which analyzes one part of an economy while holding other factors constant.
French economist Léon Walras founded general equilibrium theory in his 1874 book Elements of Pure Economics. The theory reached its modern form in the 1950s through the work of Lionel W. McKenzie, Kenneth Arrow, and Gérard Debreu.
The First Fundamental Theorem of Welfare Economics states that market equilibria are Pareto efficient. The Second Fundamental Theorem states that any Pareto efficient allocation can be supported as an equilibrium under some set of prices, meaning efficiency and equity can be addressed separately through redistributing initial endowments.
In his 1945 essay The Use of Knowledge in Society, Friedrich Hayek argued that economic knowledge is dispersed across individuals and often tacit, so it cannot be centrally aggregated as general equilibrium models assume. He proposed that market prices instead serve as decentralized information signals that coordinate an economy without centralized direction.
Economists traditionally prove existence using fixed-point theorems such as the Brouwer or Kakutani fixed-point theorem, a proof first supplied by Lionel McKenzie, Kenneth Arrow, and Gérard Debreu. Existence generally requires that consumer preferences, or feasible production sets, be convex.
Applied general equilibrium (AGE) models were pioneered by Herbert Scarf in 1967 and first implemented by John Shoven and John Whalley in 1972 and 1973, but fell out of favor in the 1980s due to high computation costs. Computable general equilibrium (CGE) models replaced them in the mid-1980s and remain the preferred method used by governments and the World Bank.