Inerter-based　dampers　(IBDs)　have　been　demonstrated　to　be　effective　in　reducing　structural　vi-bration　due　to　the　mass　amplification　and　energy　dissipation　enhancement　effects　of　inerters.　However,　previous　studies　adopted　mostly　an　idealized　linear　inerter　model,　and　the　influences　of　inerter　nonlinearities　on　the　effectiveness　of　IBDs,　especially　in　seismic-induced　vibrations　of　adjacent　structures,　have　yet　to　be　fully　studied.　To　address　this　research　gap,　a　new　nonlinear　model　is　proposed　to　characterize　the　inerter　nonlinearities,　with　the　nonlinear　parameters　identified　and　validated　through　the　mechanical　performance　tests.　An　analytical　model　is　then　established　for　the　adjacent　structures　equipped　with　an　inerter　device,　validating　its　accuracy　by　comparing　results　with　shaking　table　tests.　Finally,　the　influences　of　inerter　nonlinearities　on　three　classical　IBDs,　namely　tuned　inerter　damper　(TID),　tuned　viscous　mass　damper　(TVMD),　and　spring　dashpot　inerter　system　(SDIS),　in　reducing　the　seismic　responses　of　adjacent　structures　are　systematically　investigated.　The　experimental　and　analytical　results　demonstrate　that　inerter　nonlinearities　significantly　affect　the　relative　displacement　between　adjacent　structures,　and　the　friction　is　the　dominant　inerter　nonlinearity,　with　the　elastic　effect　being　generally　negligible.　Furthermore,　the　influences　of　nonlinearities　on　TVMD　and　TID　control　effectiveness　are　contingent　upon　external　excitations.