In　this　paper　a　Multistep　Collocation　(MC)　method　for　the　second　kind　Fredholm　integral　equations　(FIEs)　is　proposed　and　analyzed.　The　multistep　collocation　method　is　applied　to　FIE　with　smooth　kernels　under　uniform　mesh　and　weakly　singular　kernels　vertical　bar　s　-　t　vertical　bar(-alpha)　(0　＜　alpha　＜　1)　using　a　graded　mesh　then　the　same　convergence　rate　as　collocation　method　but　with　a　lower　degree　of　freedom　is　obtained.　Moreover,　in　order　to　avoid　the　round-off　errors　caused　by　graded　mesh,　a　Hybrid　Multistep　Collocation　(HMC)　method　by　combining　multistep　collocation　and　hybrid　collocation　method　is　proposed.　The　HMC　method　converges　faster　with　lower　degrees　of　freedom　and　more　efficiently　captures　the　weakly　singular　properties　by　nonpolynomial　interpolation　at　the　first　subinterval.　The　L-infinity-norm　convergence　results　are　analysed　and　proved.　Numerical　examples　are　presented　to　demonstrate　the　efficiency　of　the　proposed　methods.　(C)　2021　Elsevier　Inc.　All　rights　reserved.