We have measured the noise properties of high speed silicon and germanium avalanche photodiodes in the frequency range from a few megacycles up to several Gc, and for photocurrent multiplications up to and including breakdown. At microwave frequencies, and for voltages less than breakdown, the measured mean square noise current measured as a function of increasing multiplication M shows two distinct regions. For low values of M, the noise power is observed to increase approximately as the 2.5 power of M, in qualitative agreement with a simple model for the fluctuations in M. At high multiplications the noise is limited by the time constant associated with the multiplication. Noise measurements in this region yield a value for this time constant (typically2-4\times10^{-12}sec) which is in reasonable agreement with values obtained directly from microwave demodulation experiments. At low frequencies the noise power increases even more rapidly with current up to the breakdown region, where the noise power decreases approximately as the reciprocal of current. An important practical implication of these results is that the best signal-to-noise ratio is achieved when the multiplication is such that the multiplied shot noise is just equal to the sum of the series resistance and receiver noise. Further multiplication, in effect, only amplifies noise faster than it does signal. Thus, although avalanche multiplication does improve the detector sensitivity, the minimum detectable signal will still depend on receiver noise.