The invention concerns a dismountable folding chair or armchair structure and, in particular, a "director"-type chair, in which a fold-up seat (1), defined by pivoted elements (7) crossing one another, is hinged to side frames (2) defining the supporting legs (3), the armrests (4) and backrest elements (5) of the chair. The folding seat (1) and the side frames (2) comprise reciprocal connecting means (18,20) which may snap on and off, to stiffen and hold the chair folded down. The seat and backrest fabrics (11, 6) moreover, are removably fitted in order to allow the fabrics themselves to be changed.