To　break　through　the　topological　restriction　imposed　by　conventional　reflecting/transmitting-only　reconfigurable　intelligent　surface　(RIS)　in　covert　communication　systems,　a　simultaneously　transmitting　and　reflecting　RIS　(STAR-RIS)　is　adopted　in　this　paper.　A　transmitter　Alice　communicates　with　both　users　Willie　and　Bob,　where　Bob　is　the　covert　receiver.　Moreover,　Willie　also　plays　a　warden　seeking　to　detect　the　covert　transmission　since　it　forbids　Alice　from　illegally　using　the　communication　resources　like　energy　and　bandwidth　allocated　for　them.　To　obtain　the　maximum　covert　rate,　we　first　design　the　transmission　schemes　for　Alice　in　the　case　of　sending　and　not　sending　covert　information　and　further　derive　the　necessary　conditions　for　Alice　to　perform　covert　communication.　We　also　deduce　Willie＇s　detection　error　probability,　the　minimum　value　of　which　obtained　as　well　in　terms　of　an　optimal　detection　threshold.　Furthermore,　through　the　design　of　Alice＇s　transmit　power　for　covert　transmission　together　with　transmission　and　reflection　beamforming　at　STAR-RIS,　we　achieve　the　maximum　effective　covert　rate.　Our　numerical　results　show　the　correctness　of　the　proposed　theorems　and　indicate　that　utilizing　STAR-RIS　to　enhance　covert　communication　is　feasible　and　effective.