Recent advances in the observation and numerical modeling of magnetic flux emergence on small-scales are reviewed. The high-resolution limit of solar photospheric observations has reached scales of order 0".2, or 100-200 km, in recent years. Observations with that resolution show individual flux tubes emerging within single granules in the quiet Sun as small bipolar features of flux as low as 1016 Mx. Also, high-resolution observations of emerging ephemeral active regions have been carried out simultaneously at heights from the photosphere to the corona using different instruments in space and on the ground, and providing views of the emergence process with unprecedented detail. This paper starts with a brief review of some of the highest-resolution flux emergence observations. On the theory side, there is an increasing number of realistic numerical simulations of flux emergence that solve the equations of magnetohydrodynamics and radiation transfer. Various groups have studied different aspects of the radiation-MHD modeling of flux emergence, but their simulations in part cover the same processes. In this paper, a number of conclusions of the models are discussed with special focus on the comparison between the results of the different groups. The removal of magnetic fields from the surface is a less explored field than the inverse process, both observationally and theoretically. Yet, there is a good number of observations of flux disappearance from the photosphere and other atmospheric layers, typically in the form of cancellation of colliding flux elements of opposite polarity. On the simulation side, various numerical experiments of emerging flux regions find clear instances of flux cancellation and removal in the runs. In those cases, reconnection of field lines of opposite polarity is taking place and leads to phenomena akin to those reported in some of the observations. In this review a number of recent results from theory and observation are discussed which help understand the removal of flux from the solar atmosphere.