"They ask you [Muhammad] concerning the new moons. Say: They are but signs to mark fixed periods of time for men and for Hajj (pilgrimage)".
[Al-Baqarah (2) 189]

Fast (Muslims) when you see the crescent. If it is obscure to you, then complete thirty days in the count of Sha`ban. And break your fast when you see the crescent. If it is obscure to you, then fast thirty days.
[Al-Bukhari and Muslim]

The Messenger (salla 'Llahu `alayhi wa sallam) used to be more concerned about [determining the beginning of] Sha`ban than about other months. Then he used to fast at the sighting of [the crescent of] Ramadhan. If it were obscure, he would count thirty days [of Sha`ban] and then fast.
[Authentic; narrated by Abu Dawud and Ahmad]

Witnesses Needed to Establish the Sighting

If two just Muslim witnesses testify [that they saw it] then fast or break your fast.
[Authentic; narrated by Ahmad, an-Nassa'i and ad-Daraqutni]

People were looking out for the crescent [of Ramadhan]. I informed the Prophet (salla 'Llahu `alayhi wa sallam) that I saw it. So he fasted [on the following day] and ordered people to fast.
[Authentic; narrated by Abu Dawud and others]

Astronomical Evidence

 
We are an illiterate nation. We do not use astronomical writing or computation [in our fasting]. A month is so and so and so (and he pointed with his hands three times, folding the thumb on the third time, meaning twenty nine days) or so and so and so (and he pointed with his hands three times, meaning thirty days).
[Al- Bukhari, Muslim, Abu Dawud and An-Nassa'i]

Thus Ibn `Umar used to end his fasting with the rest of people without relying on those computations.

Anyone who rejects this would indeed overstep the ijma` (consensus) of the worthy `ulama, would be attempting to spoil the beauty and simplicity of this Din, and, worst of all, would be bluntly disobeying the Messenger (salla 'Llahu `alayhi wa sallam).

Misconception: the True Beginning of a Month

 
a. It involves casting doubts about the truthfulness of just Muslims (`udul). This is forbidden in the texts of the Qur'an and the Sunnah.

b. It conflicts with the practice of the Messenger (salla 'Llahu `alayhi wa sallam) who accepted the sighting of people without delay and without comparing it with astronomical evidence.

c. It resembles the behavior of the Jews who tightened the Laws on themselves; and Allah punished them by making them even tighter. The Messenger (salla 'Llahu `alayhi wa sallam) said:
 
Beware of [going to the extreme of] over-exaggeration in Din. Indeed, those before you were destroyed because of their over-exaggeration in Din.
[Authentic: narrated by An-Nasa'i and others]

d. It shows ignorance of when an Islamic month really begins. This point is explained by Ibn Taymiyyah (Raimahu 'Llah):
 
Many people think that once the new moon rises in the skies, that night would be the first of the month, regardless of whether people see it or not. This is not correct! It should appear to people and they should see it to start a month. That is why the Messenger (salla 'Llahu `alayhi wa sallam) said: "[True] fasting starts on the day that you start fasting ..." Which means: this is the day that you know it as being the time to fast, etc. If you did not know it, then it would carry no value for you.
[Al- Fatawa, volume 5, page 203]

We went to `Umrah and saw the crescent of Ramadhan at Dthatu `Irq. It seemed to have been two or three nights old. Later, we met Ibn `Abbas and mentioned this to him. Ibn `Abbas (Radiya 'Llahu `anhuma) informed us that Allah's Messenger (salla 'Llahu `alayhi wa sallam) said: Allah has set crescent sighting as indication of Ramadhan. Thus Ramadhan starts on the night that you see it. If it is obscure to you, then complete the count.
[Narrated by Muslim]

But what if those who claimed sighting of the new moon were lying or mistaken? As long as they are apparently acceptable and trustworthy Muslims then we must follow their testimony and fast (or end the fasting). We have absolutely no right to reject that without a definite proof (applying means which conform with the Sunnah, as outlined above.) If they happen to be indeed lying then the sin will be theirs only, and we would still be (in sha'a 'Llah) rewarded for sticking to the Sunnah.

Unification of Sighting Locations

 
[True] fasting starts on the day that you start fasting; [true] Fitr [`Id] is on the day that you end your fasting; [true] Adha [`Id] is on the day that you offer your sacrifice.
[Authentic; narrated by Abu Dawud, at-Tirmidthi and others]

Misconception: Different Matla`s

 
Kurayb reported that Umm al-Fadhl Bint Al-rith sent him on a mission to Mu`awiyah (Radiya 'Llahu `anhu) in Damascus. He accomplished his mission and was still in ash-Sham when Ramadhan started. He saw the new moon on Friday evening. He then returned to al-Madinah, arriving therein near the end of the month. He met Ibn `Abbas who asked him when was the new moon of Ramadhan sighted in ash-Sham. Kurayb said, We saw it on the night of Friday. Ibn Abbas inquired, Did you see it yourself? Kurayb replied, Yes I saw it; and People did too. Based on that, they fasted and Mu`awiyah fasted as well. Whereupon Ibn Abbas said, But we saw it on the night of Saturday; and we shall continue to fast until we complete thirty days or see it [the new moon of Shawwal]. Kurayb asked, Wouldn't you accept Mu`awiyah's sighting and fasting. Ibn Abbas answered: No! This is how Allah's Messenger commanded us.
[Narrated by Muslim]

After discussing this narration, Shayk ul-Islam Ibn Taymiyyah (Raimahu 'Llah) concluded:

 
To summarize: a person who learns about the sighting of the moon in good time to be able to utilize it for fasting, for ending his fast, or for sacrifice, he must definitely do so. The texts [of Islam] and the reports about the Salaf point to this. To limit this to a certain distance or country would contradict both the reason and the Shar` (Islamic law).
[Al-Fatawa, volume 5, page 111]

You should know that the acceptable evidence is in what Ibn Abbas (Radiya 'Llahu `anhu) reported explicitly from the Messenger (salla 'Llahu `alayhi wa sallam). It is not in his ijtihad which people tried to interpret, and to which he pointed by saying, This is how Allah's Messenger commanded us. His direct report from the Prophet (salla 'Llahu `alayhi wa sallam) is what Al-Bukhari, Muslim, and others narrated as, Do not fast until you see the crescent, and do not end your fast until you see it. This does not apply to any particular location, but is a general address to all applicable Muslims. Thus it is obviously an evidence that a sighting in one location is binding to people in other locations. This is so because when [Muslim] people in a particular location see it, this means that all Muslims have seen it. Thus what applies to them [those who saw it] would apply to others [in other locations] Despite that ash-Sham was not far enough from al-Madinah to warrant different matlas (moon rising positions), Ibn Abbas refused to follow the sighting of its people. This tells that his behavior was based on his ijtihad, and is not, therefore, an acceptable evidence [in this case].
[Nayl ul- Awtar, volume 4, page 268]

If the crescent is seen by people in one location, people in other locations must abide by their sighting. This is based on the hadiths that declare the beginning and the end of fasting at the sighting of the crescent. These hadiths address the whole Muslim Ummah: when one Muslim sees the crescent anywhere, his sighting would be a sighting for all Muslims. It would not be correct to challenge this with the hadith of Kurayb (in sahih Muslim), because Ibn `Abbas did not declare in it that the Prophet (salla 'Llahu `alayhi wa sallam) commanded them not to follow sightings of other locations.
[ar-Rawdhat un-Nadiyyah, volume 1, page 224]

In this regard also, Muhammad Nasir ud-Din al-Albani, one of the top hadith specialists, said:

 
Matla`s are relative. They have no definite boundaries which people can distinguish and to which they can refer. Most `ulama give no weight to differences in matlas. This is based on the general meaning of the authentic hadith: Fast upon seeing it; and break your fast upon seeing it. This is the truth, and none else can be accepted. One may not dispute this with the hadith of Ibn Abbas ... It may be even better to say that the hadith of Ibn Abbas concerns those who follow their locality's sighting and later during Ramadhan learn that the crescent was sighted elsewhere one day earlier. In such case, they would continue to fast with their countrymen until they complete thirty days or see their own crescent. This removes the confusion and leaves the above hadith applicable in general to all those who receive the news of seeing the crescent from any location. It is well known that this matter is very easy to accomplish nowadays.
[Tamam ul- Minnah, page 398]

Summary

 
a. The beginning of an Islamic month may be determined (for purposes of fasting, Hajj, and other Islamic occasions) only through naked-eye sighting of the new crescents.

b. Astronomical evidence may not be used to establish moon sighting or to verify or revoke the sighting of trustworthy Muslims.

c. Once the new moon is sighted anywhere on earth, people in all other places who learn about the sighting in a reasonable amount of time to be able to use it must do so.

d. A month starts when the new moon is seen regardless of whether this occurred before, after or at the actual possible sighting.