The　constrained　least　absolute　deviation　(L-1)　estimator　is　an　attractive　alternative　to　both　the　unconstrained　L-1　estimator　and　the　least-square　estimator.　This　paper　introduces　a　constrained　L-1　method　and　proposes　two　cooperative　recurrent　neural　networks　(CRNNs)　for　the　constrained　L-1　estimator.　Unlike　existing　cooperative　neural　networks,　the　proposed　two　CRNNs　have　a　novel　weighting　cooperation　scheme　to　integrate　individual　neural　network　information　automatically.　As　a　special　case,　the　proposed　continuous-time　CRNN　includes　the　existing　continuous-time　neural　network　for　unconstrained　L-1　estimator.　Compared　with　existing　continuous-time　neural　networks　for　the　constrained　L-1　estimator,　the　proposed　continuous-time　CRNN　has　a　lower　model　complexity　and　the　finite-time　convergence　to　the　exact　optimal　solution　without　any　additional　condition.　Furthermore,　compared　with　existing　numerical　algorithms　for　the　constrained　L-1　estimator,　in　addition　to　a　low　computational　complexity,　the　proposed　two　CRNNs　are　suitable　for　parallel　implementation　and　can　deal　with　the　L-1　estimation　problem　with　degeneracy.　The　proposed　two　CRNNs　are　applied　to　parameter　estimation　problems　under　non-Gaussian　noise　environments.　Simulation　results　demonstrate　that　the　proposed　CRNNs　are　indeed　effective　in　dealing　with　the　L-1　estimation　problem　with　nonunique　solutions　and　in　obtaining　a　better　solution　than　relevant　algorithms.