A powerful discovery of Joseph Fourier in the early 1800s was that certain functions could be written as an infinite sum of simple 'wave-like functions'. Such a decomposition is now known as a Fourier series, and has had wide-ranging applications across mathematics and wider science, for example in signal processing and in solving complicated differential equations. The Fourier transform describes how quickly the Fourier series converges, i.e. the decay rate of the amplitudes of the waves in the decomposition as frequency increases. One way of viewing this is that the faster the Fourier transform decays, the more wave-like the function was to begin with. Thus, some geometric information about the original object is captured by Fourier decay. This research project considers the Fourier transform of measures (mass distributions), which are analogous to functions. It is well known that the Fourier transform of a measure encodes a lot of information about its geometric structure, for example concerning its dimension, curvature properties, and arithmetic resonances. We investigate the Fourier transform, and the geometric information it encodes, in several challenging contexts. For example, we consider how it is affected when the original measure is distorted under standard geometric operations, such as projecting a measure in 2 dimensional space onto lines. Similar questions about the Hausdorff dimension of sets and measures are at the heart of geometric measure theory and we will establish Fourier analytic analogues of classical results in this direction. We also consider the Fourier transform in probabilistic settings. Brownian motion is a fundamental random process - first observed as the seemingly random path a grain of pollen follows when suspended in water - and is our archetypal example. We will consider the Fourier transform of natural (random) measures associated with Brownian motion and related processes. Finally, we will consider dynamically invariant measures, where we will use transfer operators and other tools from the thermodynamic formalism to analyse the Fourier decay.