Nowadays,　spatial　search　services　bring　unprecedented　convenience　in　peoples　daily　life　(e.g.,　location-based　services,　social　networking)　and　are　becoming　more　and　more　popular.　To　protect　the　privacy　of　outsourcing　data,　several　schemes　have　been　proposed　to　achieve　secure　search　over　encrypted　spatial　databases.　However,　most　existing　schemes　cannot　support　dynamic　updates,　which　seriously　hinders　the　practical　application　of　spatial　databases.　To　address　this　issue,　in　this　paper,　we　propose　two　novel　Spatial　Dynamic　Searchable　Encryption　(SDSE)　constructions　for　outsourcing　spatial　databases,　which　achieve　various　security　guarantees.　First,　we　present　a　basic　construction　supporting　dynamic　update　with　sub-linear　search　complexity　based　on　the　order-revealing　encryption　and　Quadtree.　Then,　to　ensure　that　updates　do　not　reveal　any　information　underlying　the　prior　modifications　beyond　some　explicit　leakage　(i.e.,　forward　security),　we　further　give　an　improved　construction　according　to　constrained　pseudo-random　functions.　Both　the　nearest　neighbor　search　and　geometric　search　are　supported　in　our　constructions,　which　meet　almost　all　spatial　search　needs.　The　experiments　using　real-world　dataset　demonstrate　that　our　constructions　are　efficient　and　feasible　in　practice.　©　Springer　Nature　Switzerland　AG　2020.