Many systems are involved in moving the limbs.
The musculo-skeletal system (muscles and bones) is typically seen as "the" system responsible for limb movement.
However, muscles won't contract unless they're innervated by the nervous system, and won't be able to perform the complex reactions needed for movement unless they're provided substances like oxygen (through blood) from the cardiovascular system. Additionally, muscles are affected by hormones and other chemicals in the body that are released by the endocrine system (they might signal growth, for example).
For further reading, you may want to look up Actin and Myosin and examine muscle contraction.
Well, technically your muscular system moves limbs.
The musculoskeletal system provides for the movement of limbs and controls facial movements.
they use their limbs and move.
Smooth muscle is responsible for the contraction that moves limbs.
The flexors are in the muscular system and their function is to bend limbs.
The muscles attached to your bones (usually in pairs) enable you to move your limbs.
The axillary artery moves oxygenated blood to the the upper limbs, axilla and thorax. :)
The main function of the peripheral nervous system is to connect the central nervous system to the limbs and organs. It helps coordinate body movements and receiving external stimuli. It is also the system that controls the "fight or flight" response.
The respiratory system is the organ system that moves air into and out of your body.