Abstract For a function f : [ 0 , 1 ] × R → R the superposition operator S f : R [ 0 , 1 ] → R [ 0 , 1 ] is defined by the formula S f ( φ ) ( t ) = f ( t , φ ( t ) ) . We study properties of operators S f in Banach spaces B V φ ( 0 , 1 ) of all real functions of bounded φ-variation on [ 0 , 1 ] for convex functions φ. We show that if an operator S f maps the space B V φ ( 0 , 1 ) into itself, then (1) it maps bounded subsets of B V φ ( 0 , 1 ) into bounded sets if additionally f is locally bounded, (2) f = f c r + f d r where the operator S f c r maps the space D ( 0 , 1 ) ∩ B V φ ( 0 , 1 ) of all right-continuous functions in B V φ ( 0 , 1 ) into itself and the operator S f d r maps the space B V φ ( 0 , 1 ) into its subset consisting of functions with countable support. Moreover we show that if an operator S f maps the space D ( 0 , 1 ) ∩ B V φ ( 0 , 1 ) into itself, then f is locally Lipschitz in the second variable uniformly with respect to the first variable.