We have learnt in the last decades that the majority of galaxies belong to high density regions interconnected in a sponge-like fashion. This large-scale structure is characterised by clusters, filaments, and walls, where most galaxies concentrate, but also under-dense regions called voids. The void regions and the galaxies within represent an ideal place for the study of galaxy formation and evolution, as they are largely unaffected by the complex physical processes that transform galaxies in high-density environments. The void galaxies may hold the key to answer current challenges to the ΛCDM paradigm as well. The CAVITY survey is a Legacy project approved by the Calar Alto Observatory to obtain spatially resolved spectroscopic information of ∼300 void galaxies in the Local Universe (0.005 < z < 0.050), covering ‑17.0 to ‑21.5 in r band absolute magnitude. It officially started in January 2021 and has been awarded 110 useful dark observing nights at the 3.5 m telescope using the PMAS spectrograph. Complementary follow-up projects, including deep optical imaging, integrated as well as resolved CO data, and integrated HI spectra, have joined the PMAS observations and naturally complete the scientific aim of characterising galaxies in cosmic voids. The extension data has been named CAVITY+. The data will be available to the whole community in different data releases, the first of which is planned for July 2024, and it will provide the community with PMAS datacubes for around 100 void galaxies through a user friendly and well documented database platform. Here, we present the survey, sample selection, data reduction, quality control schemes, science goals, and some examples of the scientific power of the CAVITY and CAVITY+ data.