Omission as Actus Reus
Omission, or the failure to act, can sometimes be grounds for criminal liability if some action is required by statute.
In almost every jurisdiction in the United States, a six foot tall expert swimmer who sits by the side of a five foot tall pool and heartily enjoys watching a four foot tall child drown is not committing any crime by his omission (assuming he isn't the lifeguard, of course). The usual reasoning behind this is that people should be punished only for deliberately adding to human misery, not for being indifferent towards it. Also, in many situations where coming to another's aid might put the rescuer at risk, there is far too much difficulty in ascertaining at what exact point the risk to the rescuer becomes too great to incur criminal liability.
However, when statute specifically requires action on the part of a citizen, saying "I didn't do any harm" is not enough. In these case, harm is done to society by not acting (in the appropriate manner, anyway). Failing to pay taxes, child support, and alimony are a few recognizable examples of omission as actus reus.
Sometimes a close relative of the victim can be held liable for the victim's death if he or she failed to act. If a father watches his child drown in a shallow pool and does nothing, he may be guilty of homicide on account of the responsibility he holds for his child's life. Similarly, a mother can be criminally liable for her child's death if she fails to administer or procure medical treatment while knowing that such an omission will lead to death.
Contractual obligation, verbal agreement, or even basic involvement can potentially lead to criminal liability for omission. If the six foot man by the pool in the first example was a lifeguard or a babysitter, he would probably be found criminally liable for the child's death. If he told the parents that he would watch the child while they stepped away and then failed to do so, he would probably be liable. Even if he negligently bumped the child into the pool and watched the child drown without acting, he could be held criminally liable for homicide on the grounds of omission.
Only when there is absolutely no connection, physical, contractual or otherwise, between the witness and the victim is omission no grounds for actus reus.