Continuous version of the Fourier series.
Can be used to represent functions that are not periodic: https://math.stackexchange.com/questions/221137/what-is-the-difference-between-fourier-series-and-fourier-transformation while the Fourier series is only for periodic functions.
Of course, every function defined on a finite line segment (i.e. a compact space).
As a more concrete example, just like the Fourier series is how you solve a the heat equation on a line segment with Dirichlet boundary conditions as shown at: Section "Solving partial differential equations with the Fourier series", the Fourier transform is what you need to solve the problem when the domain is the entire real line.