Lunar polar Hydrogen Mapper (LunaH-Map) is a 6U CubeSat that will enter a polar orbit around the Moon with a low altitude (5-12km) perilune centered on the lunar South Pole. LunaH-Map will carry two neutron spectrometers that will produce maps of near-surface hydrogen (H) at unprecedented spatial scales (~7.5 km/pixel). LunaH-Map will: 1) map H within permanently shadowed craters to determine its spatial distribution; 2) map H distributions with depth (< 1 meter); and 3) map the distribution of H in other permanently shadowed regions (PSRs) throughout the South Pole. These data will advance our understanding of lunar volatile distributions, and will inform future mission planning, specifically, landed missions and those that focus on in-situ resource utilization. LunaH-Map will produce the highest spatial resolution maps of hydrogen abundance ever acquired by a neutron detector from orbit, and will demonstrate the capability of a CubeSat platform to acquire neutron spectra. This will be achieved by orbiting with a low perilune (5km altitude) above the South Pole of the Moon, centered at -89.9°S (Shackleton Crater). The implications for this measurement are significant, as it directly informs our understanding of how lunar volatile abundances are distributed within various lunar South Pole craters and regions. Previously acquired lunar neutron maps will also benefit from an improved understanding of the spatial distribution of hydrogen within PSRs, as these data can be used to reinterpret lower spatial resolution data. Throughout the course of the 60-day science mission, LunaH-Map will acquire thermal and epithermal neutron counts over a total of 141 science orbits. Neutron count rates will be used to determine H abundances and distributions within Shackleton Crater on each orbit (60ppm - 12ppm H), and can additionally be used to map H distributions within several nearby PSRs (Haworth, Shoemaker, Faustini, Shackleton, de Gerlache, Nobile, Amundsen and Sverdrup). LunaH-Map will be capable of mapping entire PSRs with an average precision of 85 ppm - 17ppm H, and for spatial resolutions smaller than the crater diameter at an average precision of 180ppm - 36ppm H. LunaH-Map will utilize an innovative new scintillator technology called an elpasolite, specifically Cs2YLiCl6:Ce (CLYC), with high neutron detection efficiency across a wide energy range. These detectors are easily accommodated within a CubeSat due to their small form factor, as each instrument occupies just 1U of the 6U spacecraft. Two 2-cm thick (100 cm2) CLYC-based detector arrays (one covered in a thin layer of Cd) will be used to achieve neutron efficiencies equal to that of Lunar Prospector's 3He tubes. Onboard propulsion will provide V sufficient for lunar orbit insertion (LOI), all orbital maneuvers and station keeping throughout the science phase of the mission. Solar panels will generate 30 W of power. Attitude control consists of a set of 3-axis Sinclair reaction wheels. Communications use IRIS 3 X-band (MarCO CubeSat heritage) combined with Doppler for spacecraft tracking. LunaH-Map will also include a wide-angle engineering camera system (from Malin Space Science Systems (MSSS)) for outreach and non-essential engineering images. LunaH-Map development will take place over a 3-year period, and will undergo design audits to prepare the spacecraft and instruments for flight readiness. The spacecraft will be designed and built at Arizona State University (ASU). The neutron spectrometers will be designed, built and tested by Radiation Monitoring Devices (RMD) and delivered to ASU for integration into the spacecraft. The spacecraft will be delivered to the primary launch vehicle in July of 2018 and the nominal mission will begin 6 days after launch. LOI will take place 1-month after launch and separation. The Science Mission (Phase E) will take place over the next 60 days, after which the spacecraft will deorbit into a permanently shadowed crater at the South Pole.