The exosphere is the uppermost layer of the terrestrial atmosphere, mainly composed of atomic hydrogen (H) that resonantly scatters solar far-ultraviolet (FUV) photons at 121.56 nm, also referred to as Lyman-Alpha (Ly-α) emission. Analysis of this emission has been used to determine the global, three-dimensional, and time-dependent exospheric H density structure, which is essential to assess the permanent escape of H to space as well as to determine their role in governing the transient response of terrestrial plasma environment to space weather. Thus, Ly-α emission and its by-product, the H density, are highly desirable to the magnetospheric community. On the other hand, this emission can also be regarded as a significant source of contamination during studies of FUV targets such as O/B-type stars, planetary and exoplanetary atmospheres, and the circumgalactic medium, especially when observations are acquired from Earth-orbiting instruments. In this case, accurate specification of exospheric Ly-α photon flux and its subsequent removal is required by the planetary and astrophysics community studying solar/extra-solar system objects. This work introduces EXOSpy, an open-source python-based package that provides several models of terrestrial exospheric H density and calculates exospheric Ly-α emission with a high potential to contribute to investigations in both communities. We present several examples to demonstrate how EXOSpy can be used to (i) validate current and new exospheric models based on actual Ly-α radiance data, (ii) estimate exospheric contamination for a given instrument’s line-of-sight and spatial location, and (iii) provide support for new space-based FUV instrument design.