A　compound　droplet　subject　to　three-dimensional　oscillatory　shear　flow　is　studied　using　a　three-phase　lattice　Boltzmann　model.　Firstly,　focusing　on　low　values　of　capillary　number　(Ca)　where　the　compound　droplet　eventually　reaches　steady-state　oscillatory　condition,　we　study　the　effect　of　oscillatory　period,　viscosities　of　inner　and　outer　fluids　of　the　compound　droplet,　wall　confinement　and　Ca　on　the　droplet　behavior.　As　the　oscillatory　period　increases,　the　maximum　deformation　parameters　gradually　approach　the　steady-state　values　in　the　corresponding　simple　shear　flow　for　both　inner　and　outer　droplets,　and　the　compound　droplet　is　more　synchronous　with　applied　shear.　We　demonstrate　for　the　first　time　that　due　to　high　pressure　near　two　tips　inside　the　outer　droplet　the　inner　droplet　may　rotate　counterintuitively　in　a　direction　opposite　to　the　outer　one.　The　compound　droplet　undergoes　larger　deformation　when　either　droplet　is　less　viscous,　which　also　decreases　the　synchronization　between　inner　and　outer　droplets.　Increasing　confinement　ratio　not　only　promotes　the　deformations　of　both　constituent　droplets,　but　also　makes　them　more　synchronous　with　applied　shear.　It　is　also　found　that　the　maximum　deformation　parameters　of　both　droplets　increase　linearly　with　Ca　up　to　Ca=0.35　but　deviate　from　the　linearity　at　higher　Ca,　where　multipeaked　oscillations　are　observed　for　the　deformation　of　the　inner　droplet,　which　can　be　due　to　the　extensional　flow　resulting　from　the　rapid　contraction　of　the　outer　droplet.　We　then　analyze　the　breakup　behavior　of　compound　droplet　in　the　oscillatory　shear　flow　for　varying　confinement　ratios,　and　compare　the　findings　with　those　in　simple　shear　flow.　The　critical　capillary　number　for　droplet　breakup　exhibits　a　non-monotonic　behavior　with　the　confinement　ratio　in　both　shear　flows,　but　its　value　is　always　higher　in　oscillatory　shear　flow　than　in　simple　shear　flow.　As　the　confinement　ratio　increases,　in　the　case　of　oscillatory　shear　flow,　the　droplet　undergoes　a　transition　from　inner　ternary　breakup　to　inner　binary　breakup,　distinct　from　the　one　observed　in　the　case　of　simple　shear　flow.　Finally,　increasing　oscillatory　period　is　found　to　not　only　decrease　the　critical　capillary　number　but　also　change　the　mode　of　droplet　breakup.　©　2020　Elsevier　Ltd