The outburst of the unusual transient V838 Monocerotis in 2002 illuminated surrounding interstellar dust, producing the most spectacular light echoes in astronomical history. The echoes remained visible for several years. Because of the light-echo geometry, each imaging observation corresponds uniquely to a well-defined paraboloidal region of space illuminated by the outburst. It is therefore possible to reconstruct the true three-dimensional (3D) distribution of dust around V838 Mon from a series of observations of its light echoes. We obtained images of the echoes using the Hubble Space Telescope (HST) from 2002 to 2011, as well as complementary ground-based data, in order to monitor the evolution of the light echoes at regular time intervals. Based on these images, we reconstruct and visualize the 3D dust density distribution around V838 Mon. Such mapping of the structure helps determine whether the illuminated material represents an outflow from the progenitor of V838 Mon, or pre-existing interstellar dust. A subset of the HST observations, acquired with a high temporal cadence, paves the way to a detailed study of a slab of the dust cloud around V838 Mon on spatial scales as small as ~0.0015 pc or ~310 AU.