Permanently cross-linked materials have outstanding mechanical properties and solvent resistance, but they cannot be processed and reshaped once synthesized. Non-cross-linked polymers and those with reversible cross-links are processable, but they are soluble. We designed epoxy networks that can rearrange their topology by exchange reactions without depolymerization and showed that they are insoluble and processable. Unlike organic compounds and polymers whose viscosity varies abruptly near the glass transition, these networks show Arrhenius-like gradual viscosity variations like those of vitreous silica. Like silica, the materials can be wrought and welded to make complex objects by local heating without the use of molds. The concept of a glass made by reversible topology freezing in epoxy networks can be readily scaled up for applications and generalized to other chemistries.