CAFE (Continuum And Feature Extraction) fits JWST IFU data; the code is a Python version of the original CAFE IDL software for fitting Spitzer/IRS spectra. The code contains two main tools: (1) the CAFE Region Extraction Tool Automaton (CRETA) and (2) the CAFE spectral fitting tool, or fitter. CRETA performs single-position and full-grid extractions from JWST IFU datasets; that is, from pipeline-processed cubes obtained with the NIRSpec IFU and MIRI MRS instruments. The CAFE fitter uses the spectra extracted by CRETA (or spectra provided by the user) and performs a spectral decomposition of the continuum emission (stellar and/or dust), as well as of a variety of common spectral features (in emission and absorption) present in the near- and mid-IR spectra of galaxies, including prominent, broad emission from small grains and molecules such as Polycyclic Aromatic Hydrocarbons (PAHs). The full dust treatment (size and composition) performed by CAFE allows the dust continuum model components to fit not only spectra from typical star-forming galaxies, but also those from more extreme, heavily dust-obscured starburst galaxies, such as luminous infrared galaxies (LIRGs and ULIRGs), active galactic nuclei (AGN), or very luminous quasars.