Formaldehyde-free binder compositions are described that include an aldehyde or ketone, an organic anhydride, an alkanol amine, and a nitrogen-containing salt of an inorganic acid. The binder compositions may be applied to fibers, such as glass fibers, to make formaldehyde-free, fiber-reinforced composites. Methods of making fiber-reinforced composites are also described, where such methods may include mixing an alkanol amine with an organic anhydride to make a first mixture, and adding a reducing sugar to the first mixture to make a second mixture. A nitrogen-containing salt may be added to the second mixture to make a binder composition, which may be applied to fibers to form a binder-fiber amalgam. The amalgam may be heated to cure the binder composition and form the fiber-reinforced composite.