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$\texttt{Eureka!}$ is a data reduction and analysis pipeline for exoplanet time-series observations, with a particular focus on JWST data. Over the next 1-2 decades, JWST will pursue four main science themes: Early Universe, Galaxies Over Time, Star Lifecycle, and Other Worlds. Our focus is on providing the astronomy community with an open source tool for the reduction and analysis of time-series observations of exoplanets in pursuit of the fourth of these themes, Other Worlds. The goal of $\texttt{Eureka!}$ is to provide an end-to-end pipeline that starts with uncalibrated FITS files and ultimately yields precise exoplanet spectra. The pipeline has a modular structure with six stages, and each stage uses a "Eureka! Control File" (ECF) to allow for easy control of the pipeline's behavior. Stage 5 also uses a "Eureka! Parameter File" (EPF) to control the fitted parameters. We provide template ECFs for the MIRI, NIRCam, NIRISS, and NIRSpec instruments on JWST and the WFC3 instrument on the Hubble Space Telescope (HST). These templates give users a good starting point for their analyses, but $\texttt{Eureka!}$ is not intended to be used as a black box tool, and users should expect to fine-tune some settings for each observation in order to achieve optimal results. At each stage, the pipeline creates intermediate figures and outputs that allow users to compare $\texttt{Eureka!}$'s performance using different parameter settings or to compare $\texttt{Eureka!}$ with an independent pipeline. The ECF used to run each stage is also copied into the output folder from each stage to enhance reproducibility. Finally, while $\texttt{Eureka!}$ has been optimized for exoplanet observations (especially the latter stages of the code), much of the core functionality could also be repurposed for JWST time-series observations in other research domains thanks to $\texttt{Eureka!}$'s modularity.