Random fluctuations in the amplitude of an electric current were originally discovered in two different circumstances: (a) the fluctuation in the current through a thermionic valve, which is known as “shot noise,” and (b) the voltage fluctuation at the terminals of a resistor, which is known as “Johnson noise.” The theory of these two types of fluctuation is presented in outline form; and it is shown how these two aspects of the fundamentally discrete nature of the flow of electric current, when it is a stream of electrons, can be unified into a single theory of current fluctuations due to random motions of the electrons. The theory is applied also to the controversial case of a space-charge-limited thermionic valve. Subsidiary factors, such as partition of current in a screen-grid valve and the behaviour of secondary-emission amplifiers, are of great technical importance, and methods of estimating the magnitude of their effects are described. The equivalent temperature of the radiation resistance of an aerial is also considered.