A spark-ignition direct injection engine is provided. The engine includes an engine body, a fuel injection valve, a fuel pressure setting mechanism, an ignition plug, and a controller. The controller operates the engine to perform compression-ignition combustion within a first operating range, and controls the ignition plug to operate the engine to perform spark-ignition combustion within a second operating range. Within a specific part of the first range, the controller sets the fuel pressure to 30 MPa or above, and retards the compression ignition to after a compression top dead center by controlling the injection valve to inject fuel into a cylinder in a period from a late stage of compression stroke to an early stage of expansion stroke. Below the specific part, the controller controls the fuel injection valve to inject the fuel into the cylinder in a period from intake stroke to a mid-stage of the compression stroke.