The neutron densities in atomic nuclei are notoriously difficult to observe with high precision: the standard tool of electromagnetic interactions which has been used to map out the nuclear charge distributions simply doesn't see them. In fact, it has only recently been experimentally confirmed that the neutron-rich lead nucleus even has a neutron skin, and is only a fraction of a neutron radius thick. Encoded in these distributions is a wealth of important information about how the strong nuclear force builds systems where the number of protons and neutrons are unequal. This information has bearing not only for our understanding of asymmetric nuclei, but also in the construction of extreme systems like neutron stars. Fortunately, nature gives us a novel way to image this side of the nucleus: through fundamental weak force interactions, which interact primarily to neutrons rather than protons. In this colloquium I will discuss why these neutron distributions play an important part in our understanding of nuclear physics and astrophysics, how one images such tiny systems with electron beams, and the recent and upcoming experimental efforts for such measurements.