We present a complete exploration of the different fragmentation mechanisms of furan (C4H4O) operating at low and high energies. Three different theoretical approaches are combined to determine the structure of all possible reaction intermediates, many of them not described in previous studies, and a large number of pathways involving three types of fundamental elementary mechanisms: isomerization, fragmentation, and H/H2 loss processes (this last one was not yet explored). Our results are compared with the existing experimental and theoretical investigations for furan fragmentation. At low energies the first processes to appear are isomerization, which always implies the breaking of one C-O bond and one or several hydrogen transfers; at intermediate energies the fragmentation of the molecular skeleton becomes the most relevant mechanism; and H/H2 loss is the dominant processes at high energy. However, the three mechanisms are active in very wide energy ranges and, therefore, at most energies there is a competition among them.