This　paper　addresses　the　problem　of　multistability　of　competitive　neural　networks　with　nonlinear,　non-monotonic　piecewise　activation　functions　and　time-varying　delays.　Several　sufficient　conditions　are　proposed　to　guarantee　the　existence　of　(2K+1)n$$　{\left(2K+1\right)}＜^＞n　$$　equilibrium　points　and　the　locally　exponential　stability　of　(K+1)n$$　{\left(K+1\right)}＜^＞n　$$　equilibrium　points,　where　K$$　K　$$　is　a　positive　integer　and　determined　by　the　property　of　activation　functions　and　the　parameters　of　neural　networks.　The　quantitative　relationship　between　the　equilibrium　points　of　the　system　and　the　zero　roots　of　the　bounding　functions　is　given.　In　addition,　the　attraction　basins　of　the　exponentially　stable　equilibrium　points　are　obtained.　Finally,　a　numerical　simulation　is　given　to　illustrate　the　effectiveness　of　the　obtained　results.