A convertible sofabed is a sofa when closed in a sitting position and a bed when opened in a sleeping position. A retractable support structure can be telescoped from the sofa to form, in conjunction with the sofa, a bed that is longer than the depth of the sofa in its sitting position. The mattress in the sleeping position is folded in three portions, a fixed portion, a central portion and an outer portion. The outer portion forms a sofaback in the sitting position and the central portion folds to lie on top of the fixed portion to form the seat of the sofabed. The support structure includes curved legs that travel easily over the floor without causing damage.