Narrated Ibn Abbas (radiallaahu `anhu):
When Allah’s Apostle (sallallaahu `alayhi wasallam) and his companions came to Mecca, the pagans circulated the news that a group of people were coming to them and they had been weakened by the Fever of Yathrib (Medina). So the Prophet ordered his companions to do Ramal in the first three rounds of Tawaf of the Ka’ba and to walk between the two corners (The Black Stone and Yemenite corner). The Prophet (sallallaahu `alayhi wasallam) did not order them to do Ramal in all the rounds of Tawaf out of pity for them.
Narrated Salim that his father said:
I saw Allah’s Apostle (sallallaahu `alayhi wasallam) arriving at Mecca; he kissed the Black Stone Corner first while doing Tawaf and did ramal in the first three rounds of the seven rounds (of Tawaf).
Narrated Abdullah bin Umar (radiallaahu `anhu):
The Prophet (sallallaahu `alayhi wasallam) did Ramal in (first) three rounds (of Tawaf), and walked in the remaining four, in Hajj and Umra.
Narrated Zaid bin Aslam from his father who said:
“Umar bin Al-Khattab (radiallaahu `anhu) addressed the Corner (Black Stone) saying, ‘By Allah! I know that you are a stone and can neither benefit nor harm. Had I not seen the Prophet touching (and kissing) you, I would never have touched (and kissed) you.’ Then he kissed it and said, ‘There is no reason for us to do Ramal (in Tawaf) except that we wanted to show off before the pagans, and now Allah has destroyed them.’ ‘Umar (radiallaahu `anhu) added, ‘(Nevertheless), the Prophet (sallallaahu `alayhi wasallam) did that and we do not want to leave it (i.e. Ramal).’
Ibn ‘Umar (radiallaahu `anhu) said, “I have never missed the touching of these two stones of Ka’ba (the Black Stone and the Yemenite Corner) both in the presence and the absence of crowds, since I saw the Prophet (sallallaahu `alayhi wasallam) touching them.” I asked Nafi’: “Did Ibn ‘Umar (radiallaahu `anhu) use to walk between the two Corners?” Nafi’ replied, “He used to walk in order that it might be easy for him to touch it (the Corner Stone).”
It is a Sunnah to trot in the first three circuits of tawaaf and it is Sunnah to jog in saa‘i between the two markers; these are two Sunnahs that apply to men, not women. One should walk fast between the two markers in all laps of saa’i, from al-Safa to al-Marwah and from al-Marwah to al-Safa.
Ibn Qudaamah said in al-Mughni (10/236), when describing saa’i between al-Safa and al-Marwah:
It means coming down from al-Safa and walking until you are level with the marker, which is the green mark that is attached to the corner of the mosque. When you are about six cubits away from it, you should walk quickly until you come in line with the other marker, then you should stop trotting and walk (normally) until you come to al-Marwah, then stand to face the qiblah and say du’aa’ as you did at al-Safa. … Then come down and walk in the place for walking, and trot in the place for trotting. End quote.
The scholars of the Standing Committee for Issuing Fatwas were asked about that and said:
Yes, it is the same when going from al-Marwah to al-Safa as when going from al-Safa to al-Marwah, because the Prophet (peace and blessings of Allaah be upon him) did that, as is proven in the saheeh ahaadeeth.
Ibn al-Mundhir said:
The scholars are unanimously agreed that women do not have to trot around the Ka’bah or between al-Safa and al-Marwah, and they do not have to uncover one shoulder, because the purpose behind these two actions is to show strength, and women are not included in that because the basic principle for women is to conceal themselves, but in trotting and baring one shoulder they would be uncovering themselves. End quote.
[Taken from IslamQA]