This chapter treats in detail the application of the Fourier transform in linear filtering as an important part in Electrical Engineering and other fields. The first part is devoted to the continuous case, e.g., analog linear circuits. A fundamental theorem on the convolution representation of causal translation-invariant linear systems (LTI systems) is the starting point. All systems are considered as operators on a signal space of distributions. It is shown how different types of linear filters can be mathematically designed. The examples are lowpass, bandpass, allpass, and bandstop filters with their mathematical representation and with their realization by standard circuits. Conditions for stability of linear filters by their frequency response are deduced. The second part on discrete linear filters starts again with a theorem that causal systems have a convolution representation by their impulse response. Counterexamples are given, if causality is missing and the input signal space is the space of bounded discrete signals. The z-transform is introduced for a treatment of discrete LTI systems with their transfer functions. Invertibility and design of causal linear phase FIR filters is studied. Causal IIR filters are calculated with the bilinear transform. All topics are completed with examples and exercises.