In　this　paper,　rich　dynamics　and　complex　bifurcations　of　an　SIR　epidemic　model　with　saturated　incidence　rate　and　strong　Allee　effect　are　investigated.　First,　the　existence　of　disease-free　and　endemic　equilibria　is　explored,　and　we　prove　that　the　system　has　at　most　three　positive　equilibria,　which　exhibit　different　types　such　as　hyperbolic　saddle　and　node,　degenerate　unstable　saddle　(node)　of　codimension　2,　degenerate　saddle-node　of　codimension　3　at　disease-free　equilibria,　and　cusp,　focus,　and　elliptic　types　Bogdanov-Takens　singularities　of　codimension　3　at　endemic　equilibria.　Second,　bifurcation　analysis　at　these　equilibria　are　investigated,　and　it　is　found　that　the　system　undergoes　a　series　of　bifurcations,　including　transcritical,　saddle　node,　Hopf,　degenerate　Hopf,　homoclinic,　cusp　type　Bogdanov-Takens　of　codimensional　2,　and　focus　and　elliptic　type　Bogdanov-Takens　bifurcation　of　codimension　3　which　are　composed　of　some　bifurcations　with　lower　codimension.　The　system　shows　very　rich　dynamics　such　as　the　coexistence　of　multiple　periodic　orbits　and　homoclinic　loops.　Finally,　numerical　simulations　are　conducted　on　the　theoretical　results.