On the Airbus, there are 3 hydraulic systems, Green, Yellow and Blue. If a hydraulic system fails, the PTU, Power Transfer Unit (the thing which makes a "bark" noise) will keep all 3 hydraulic systems pressurised. All 3 hydraulic systems can extend the flaps and slats.
If, however, 2 hydraulic systems have lost hydraulic fluid and cannot be pressurised, then things get messy. As mentioned, various flaps and slats use one of the 3 hydraulic systems. If Green fails and the hydraulic fluid leaks out, only some flaps and slats stop working. But if both Green and Yellow fail, or Green and Blue or Yellow and Blue fail, then flap failure can occur.
In such a case, the aircraft is limited to Cat 1 landings. The pilots must then reference the QRH for what their approach speed in failure should be and they will also see what their landing distance will be.
Ultimately, the aircraft will need to use more reverse thrust, and depending on the hydraulic failure, will either have to brake more or use spoilers more. Since the Airbus has brake fans, overheating of the brakes is less of an issue.
Also, if there is a dual hydraulic failure, on flare, the aircraft will enter Direct Law. At all other points the aircraft should be able to stay in Normal Law.