The last step in the process of becoming a naturalized U.S. citizen is to attend a swearing-in ceremony and take an oath of allegiance to the United States. The full text of this oath is as follows:
"I hereby declare, on oath, that I absolutely and entirely renounce and abjure all allegiance and fidelity to any foreign prince, potentate, state, or sovereignty, of whom or which I have heretofore been a subject or citizen; that I will support and defend the Constitution and laws of the United States of America against all enemies, foreign and domestic; that I will bear true faith and allegiance to the same; that I will bear arms on behalf of the United States when required by the law; that I will perform noncombatant service in the Armed Forces of the United States when required by the law; that I will perform work of national importance under civilian direction when required by the law; and that I take this obligation freely, without any mental reservation or purpose of evasion; so help me God."
Notice that one of the lines says "I will bear arms on behalf of the United States when required by law." For many, however, including members of religious communities with a history of pacifism (Quakers, Buddhists, and Mennonites) as well as conscientious objectors, such an act would be contrary to their most fundamental beliefs.
In recognition of this fact, U.S. immigration policy has long allowed such persons to request that they be allowed to take a modified oath of allegiance, in which they leave out the language about bearing arms (though they must still include the portion that promises performing work of national importance under civilian direction). As explained in Nolo's book, Becoming a U.S. Citizen, the best way to request this is to briefly explain your status in your application cover letter and also include a sworn statement detailing your beliefs and request.
Nevertheless, in the past, a perception developed that this request would mainly be granted to people who could actually prove a religious affiliation. In response, U.S. Citizenship and Immigration Services (USCIS) recently issued a Policy Alert. This clarifies this long-held policy, explaining that applicants:
This does not represent any huge change in policy -- applicants must still formally request this oath modification, and must convince USCIS that their request is a sincere one that should be honored. However, it does put conscientious objectors who can't show any formal religious affiliation on a more equal footing with those who can.