In the criminal law, how does criminal liability for omissions arise?

For criminal liability to arise, there must be both the actus reus (guilty act) and the mens rea (guilty mind). If the actus reus and the mens rea are established and there is no valid defence, then the defendant will be guilty. Within the actus reus, there must be both a voluntary act and a consequent result. Omissions however, act as an exception to this voluntary act requirement.
Indeed, an omission is a failure to act where there was a duty to do so or where it is reasonble to expect a person to act (based on a relationship or role) - example: parents would be convicted of murder if they fail to feed their starving child. Thus, sometimes omissions can give rise to criminal liability with the failure to act = actus reus of a crime.