An applicator for applying a cosmetic composition to the eyelashes or eyebrows, comprising: - a core having a longitudinal axis, the core being twisted, - at least one element attached to the core, each element comprising a central part fitted onto the core and application members carried by the central part, each application member having at least one protruding relief for catching the eyelashes that is formed by: ∘ an enlarged head (40; 60; 182; 203; 222; 232; 301), each application member being flattened in a flattening plane common to all the application members of the element, or ∘ a bent portion (164; 152, 151), each application member having a strand of material which describes at least one slot, the application member having a leg which is attached radially to the central part, a first elbow substantially at a right angle, a second elbow substantially at a right angle toward the outside, a third elbow substantially at a right angle toward an elongation axis of the leg, and a fourth elbow substantially at a right angle toward the outside, a distal portion of the application member being straight and eccentric with respect to the elongation axis of the leg, or each application member comprising at its end a hook to catch the eyelashes, or ∘ a toothed portion, each application member comprising a protruding relief defined by a tooth formed at the meeting point of two pointed arches (82, 83) that are concave toward the outside, each element being threaded onto the core along its longitudinal axis before the core is twisted, the core and the element being mechanically connected in order to substantially prevent the element from pivoting freely on the core. - a core having a longitudinal axis, the core being twisted, - at least one element attached to the core, each element comprising a central part fitted onto the core and application members carried by the central part, each application member having at least one protruding relief for catching the eyelashes that is formed by: ∘ an enlarged head (40; 60; 182; 203; 222; 232; 301), each application member being flattened in a flattening plane common to all the application members of the element, or ∘ a bent portion (164; 152, 151), each application member having a strand of material which describes at least one slot, the application member having a leg which is attached radially to the central part, a first elbow substantially at a right angle, a second elbow substantially at a right angle toward the outside, a third elbow substantially at a right angle toward an elongation axis of the leg, and a fourth elbow substantially at a right angle toward the outside, a distal portion of the application member being straight and eccentric with respect to the elongation axis of the leg, or each application member comprising at its end a hook to catch the eyelashes, or ∘ a toothed portion, each application member comprising a protruding relief defined by a tooth formed at the meeting point of two pointed arches (82, 83) that are concave toward the outside, each element being threaded onto the core along its longitudinal axis before the core is twisted, the core and the element being mechanically connected in order to substantially prevent the element from pivoting freely on the core. ∘ an enlarged head (40; 60; 182; 203; 222; 232; 301), each application member being flattened in a flattening plane common to all the application members of the element, or ∘ a bent portion (164; 152, 151), each application member having a strand of material which describes at least one slot, the application member having a leg which is attached radially to the central part, a first elbow substantially at a right angle, a second elbow substantially at a right angle toward the outside, a third elbow substantially at a right angle toward an elongation axis of the leg, and a fourth elbow substantially at a right angle toward the outside, a distal portion of the application member being straight and eccentric with respect to the elongation axis of the leg, or each application member comprising at its end a hook to catch the eyelashes, or ∘ a toothed portion, each application member comprising a protruding relief defined by a tooth formed at the meeting point of two pointed arches (82, 83) that are concave toward the outside,