A number of approaches have been employed for the design and analysis of two-fluid heat exchangers. These approaches include the effectiveness (ε or P)-number of transfer units (NTU) method, the mean temperature difference method, and the ones using the Mueller and Roetzel charts. A comprehensive review is presented of solution methods for obtaining effectiveness-NTU relationships for simple and complex heat exchangers and heat exchanger assemblies. These include analytical methods for obtaining exact solutions, approximate (including semi-numerical and analog) methods, numerical methods, matrix formalism, and additional methods based on the properties of the exchanger configurations. Additional details that pertain to the Laplace transforms and matrix formalism that includes the chain rule methodology are discussed. A step-by-step procedure that illustrates the chain rule methodology is presented. This review article includes 73 references.