The Doppler weather radar and its signals are examined from elementary considerations to show the origin and development of useful weather echo properties such as signal-to-noise ratio (SNR), range correlation, signal statistics, etc. We present a form of the weather radar equation which explicitly shows the echo power loss due to finite receiver bandwidth and how it is related to the range weighting function. Echoes at adjacent range samples have a correlation that depends on receiver bandwidth-transmitter pulsewidth product as well as sample spacing. Stochastic Bragg scatter from clouds is examined, but experimental work is required to determine if this echo power is larger than incoherently scattered power. Section III presents the relation between Doppler power spectrum and the distribution of reflectivity and velocity within a resolution volume. A new formula that relates spectrum width to the shear of radial velocities as well as turbulence, signal decorrelation from antenna rotation, and signal processing biases is presented. The estimation of power spectral moments is reviewed and properties of the most commonly used algorithms are discussed. Section V highlights some of the considerations that need to be made for Doppler radar observation of severe thunderstorms. Echo coherency is shown to limit the pulsed Doppler radar's unambiguous range and velocity measurements. Single and dual Doppler-radar techniques for wind measurements are reviewed. Observations of thunderstorms show tornado cyclones, and clear air measurements in the boundary layer reveal turbulence and waves.