During space missions, astronauts are exposed to a stream of energetic and highly ionizing radiation particles that can suppress immune system function, increase cancer risks and even induce acute radiation syndrome if the exposure is large enough. As human exploration goals shift from missions in low-Earth orbit (LEO) to long-duration interplanetary missions, radiation protection remains one of the key technological issues that must be resolved. In this work, we introduce the NEUtron DOSimetry & Exploration (NEUDOSE) CubeSat mission, which will provide new measurements of dose and space radiation quality factors to improve the accuracy of cancer risk projections for current and future space missions. The primary objective of the NEUDOSE CubeSat is to map the in situ lineal energy spectra produced by charged particles and neutrons in LEO where most of the preparatory activities for future interplanetary missions are currently taking place. To perform these measurements, the NEUDOSE CubeSat is equipped with the Charged & Neutral Particle Tissue Equivalent Proportional Counter (CNP-TEPC), an advanced radiation monitoring instrument that uses active coincidence techniques to separate the interactions of charged particles and neutrons in real time. The NEUDOSE CubeSat, currently under development at McMaster University, provides a modern approach to test the CNP-TEPC instrument directly in the unique environment of outer space while simultaneously collecting new georeferenced lineal energy spectra of the radiation environment in LEO.