This chapter provides a comprehensive look at the discrete Fourier transform (DFT) and its broad applications, from signal processing and communications to medical imaging and economic analysis. It starts by introducing the transition from continuous to discrete data, presenting the DFT with a focus on its properties, including linearity, circular shifts, and symmetry, which underpin digital signal analysis. The chapter then explores the fast Fourier transform (FFT), highlighting the efficiency of the Cooley-Tukey algorithm and mixed radix methods for faster DFT computation. Spectrum analysis, sampling, and interpolation techniques are also discussed, along with windowing functions to reduce spectral leakage in applied contexts. A section on noise characterization explains various noise types, clarifying the challenges of maintaining signal fidelity. The chapter wraps up with insights into quantization and other practical considerations in DFT, equipping readers to handle complex digital systems in real-world, noisy environments.