This paper explores the potential role of recurrent neural networks for solving linear programs. The emphases of the paper are on analyzing the asymptotic properties of recurrent neural networks that are relevant to linear programming and on developing general principles for designing such neural networks. A class of recurrent neural networks with monotonically increasing penalty variables is presented for solving linear programming problems. The proposed recurrent neural networks are asymptotically stable and able to generate optimal solutions to linear programming problems. The asymptotic properties of the proposed recurrent neural networks for linear programming are analyzed theoretically, and the design principles for synthesizing the recurrent networks are discussed based on the results of analysis. Some illustrative examples are also presented to demonstrate the performance behavior and operational characteristics of the recurrent neural networks.