This chapter provides a systematic discussion on the application of cost functions and their extended forms in efficiency and productivity analysis, aiming to address common issues in production function estimation, such as endogeneity, time lags, and limited data availability. It begins by outlining the fundamental principles of cost functions as functions of factor prices and their dual relationship with production functions, highlighting their advantages and applicability in measuring total factor productivity (TFP) changes. Compared with production functions, cost functions can largely mitigate endogeneity arising from the bidirectional causality between inputs and outputs, while retaining compatibility with various theoretical derivations of production functions, thus serving as a powerful tool for efficiency analysis. The chapter then examines the use of restricted cost functions when certain factor price data are unavailable, allowing estimation of the cost–output relationship under partial price information. This approach is further applied to estimate factor demand and substitution elasticities, revealing the degree of input substitutability and structural adjustment potential; to measure and decompose productivity growth into components such as technological progress, technical efficiency, and scale efficiency; and to assess market power in order to evaluate firms’ pricing ability and competition levels. Finally, the discussion extends to generalized cost functions, which incorporate distortions in factor prices and imperfect market conditions, thereby capturing producer behavior and resource allocation efficiency more accurately in environments with market frictions, institutional constraints, or policy interventions. Overall, this chapter offers a comprehensive methodological framework—from traditional cost functions to restricted and generalized cost functions—integrated with analytical tools for factor substitutability, market power, and productivity decomposition, thereby laying a solid econometric foundation for subsequent empirical research.