The e-ASTROGAM is a gamma-ray space mission to be proposed as the M5 Medium-size mission of the European Space Agency. It is dedicated to the observation of the Universe with unprecedented sensitivity in the energy range 0.2 - 100 MeV, extending up to GeV energies, together with a groundbreaking polarization capability. It is designed to substantially improve the COMPTEL and Fermi sensitivities in the MeV-GeV energy range and to open new windows of opportunity for astrophysical and fundamental physics space research. e-ASTROGAM will operate as an open astronomical observatory, with a core science focused on (1) the activity from extreme particle accelerators, including gamma-ray bursts and active galactic nuclei and the link of jet astrophysics to the new astronomy of gravitational waves, neutrinos, ultra-high energy cosmic rays, (2) the high-energy mysteries of the Galactic center and inner Galaxy, including the activity of the supermassive black hole, the Fermi Bubbles, the origin of the Galactic positrons, and the search for dark matter signatures in a new energy window; (3) nucleosynthesis and chemical evolution, including the life cycle of elements produced by supernovae in the Milky Way and the Local Group of galaxies. e-ASTROGAM will be ideal for the study of high-energy sources in general, including pulsars and pulsar wind nebulae, accreting neutron stars and black holes, novae, supernova remnants, and magnetars. And it will also provide important contributions to solar and terrestrial physics. The e-ASTROGAM telescope is optimized for the simultaneous detection of Compton and pair-producing gamma-ray events over a large spectral band. It is based on a very high technology readiness level for all subsystems and includes many innovative features for the detectors and associated electronics.