"May Allah bless him and grant him peace.": (Arabic: صلى الله عليه وسلم ṣall Allahu ʿalayhi wa sallam - S.A.W. or SAAW) - this expression follows specifically after saying the name of the last prophet of Islam, Muhammad.
In Arabic these salutations are called salawāt, and are abbreviated by some with the use of SAW (in accordance with the Arabic words sallallahou alayhi wasallam) or PBUH (which stands for Peace be upon him in English). However, this practice is considered to be controversial among senior Islamic scholars who disagree with this use on the basis that it demonstrates a lack of respect and laziness.
The phrase is also encoded as a ligature at Unicode codepoint U+FDFAﷺ.
Qur'anic evidence for asking the blessings on Muhammad
"The Messenger of Allah said, May he be humiliated, the man in whose presence I am mentioned and he does not send Salah upon me; may he be humiliated, the man who sees the month of Ramadan come and go, and he is not forgiven; may he be humiliated, the man whose parents live to old age and they do not cause him to be granted admittance to Paradise."
Al-Tirmidhi said that this hadith was, "Hasan gharib" (Good but only reported once).
"One morning the Messenger of Allah was in a cheerful mood and looked happy. They said, 'Oh Messenger of Allah, this morning you are in a cheerful mood and look happy.' He said, Of course, just now someone [an angel] came to me from my Lord [Allah] and said, 'Whoever among your Ummah sends Salah upon you, Allah will record for him ten good deeds and will erase for him ten evil deeds, and will raise his status by ten degrees, and will return his greeting with something similar to it.'"
—Abu Talha ibn Thabit
The isnad (chain of narrators) of this hadith is good.
A supplication remains suspended between heaven and earth and does not ascend any further until a person sends Salah on me. Do not treat me like a spare water container, send Salah upon me at the beginning of your supplication, at the end, and in the middle.
Commentary regarding abbreviating the Salah on Muhammad
"As it is prescribed to send blessings upon the Prophet (peace and blessings of Allah be upon him) in prayer when saying the tashahhud, and it is prescribed when giving khutbahs, saying Du’a and praying for forgiveness, and after the Adhan, and when entering and exiting the mosque, and when mentioning him in other circumstances, so it is more important to do so when writing his name in a book, letter, article and so on. So it is prescribed to write the blessing in full so as to fulfil the command that Allah has given to Muslims, and so that the reader will remember to say the blessing when he reads it. So one should not write the blessing on the Prophet (peace and blessings of Allah be upon him) in short form such as writing (S) or (SAWS) etc, or other forms that some writers use, because that is going against the command of Allah in His Book, where He says (interpretation of the meaning):
"Send your Salaah on (ask Allah to bless) him (Muhammad), and (you should) greet (salute) him with the Islamic way of greeting (salutation, i.e. As‑Salaamu ‘Alaykum)"
And that (writing it in abbreviated form) does not serve that purpose and is devoid of the virtue of writing 'salla Allaahu ‘alayhi wa salaam (May Allah send blessings and peace upon him)' in full. Moreover the reader may not take notice of it and may not understand what is meant by it. It should also be noted that the symbol used for it is regarded as disapproved by the scholars, who warned against it."
When mentioning sahaba (the companions of Muhammad), radhi Allahu anhu (for males) and radhi Allahu anha (for females) are used by Sunnis; they mean may Allah be pleased with him or her respectively. The phrase is sometimes also used after mentioning other names including that of Jesus and Moses, but the term عليه سلام aleyhi salaam, "On him be peace" is more common. See for example the letter from Mahmoud Ahmadinejad, President of Iran, to George W. Bush: "Can one be a follower of Jesus Christ (PBUH), the great Messenger of God, Feel obliged to respect human rights ..."