Turbos must have wastegates so that the amount of boost pressure delivered can be controlled. If the wastegate is open it allows exhaust gas to bypass the turbine (hot side of the turbo) thus reducing the turbo speed.
There are internal wastegates (built into turbo with an external actuator) and external wastegates (which often give better boost control).
There are pretty nasty consequences for having a wastegate that cannot handle the boost.
A dump valve releases excess pressure in the boost pipes when you back off the throttle. Without one you get a pressure wave that bounces off the throttle butterfly and reverses up the boost pipes and rushes back through the compressor side of the turbo. This makes the "chatter noise" (and btw it's nothing to do with the wastegate).
Two basic types of DV are atmospheric and recirculating. Atmospheric dumps to atmosphere (classic tsssh noise) recirc will dump into the inlet side of the compressor (this is what OEM turbo engines do to reduce noise).
A turbo doesn't neccesarily need a wastegate if it is sized to reach peak turbo speed at peak engine speed. This system has been used by Saab on a production engine with the V6 Asymetricaly turbo charged engine used in the 9-5. The downside to this is that there is no way effectively ramp up turbo speed to increase delivered airmass at low engine speeds.
Also modern VNT/VGT turbos do not use wastegates and instead use pneumatic or electric actuators to control the size/shape of the nozzle or delivery vane position to control turbo speed.
If a conventional wastegate is too small to flow sufficent exhaust gas to maintain a set boost pressure then the effect is boost creep, which will result in an increase in turbospeed above the desired level. This isn't neccesarily a bad thing if the resulting turbo speed is within limits and fueled for.
Cheers
M