And tell the believing women to lower their gaze (from looking at forbidden things), and protect their private parts (from illegal sexual acts, etc.)…
[Qur’aan 24: Portion of Verse 31]
Based on this a large number of scholars held the opinion that it is not permissible for a woman to look at a man who is a stranger with or without desire. 
Other scholars have said that it is permissible for her to look at males who are strangers without desire as is established in the following authentic hadith:
Aisha said, "While the Ethiopians were playing with their small spears, Allah's Messenger (sallallaahu alayi wa sallam) screened me behind him and I watched (that display) and kept on watching till I left on my own." So you may estimate of what age a little girl may listen to amusement.
Mujaahid said: When a woman advances the devil sits on her head and beautifies her for the one who look at her. When she leaves he sits on her backside and beautifies it for the one who looks at her. 
al-Qurtubi says that some scholars have used the hadiith which follows to say that it is permissible for a woman to see of a man who is a stranger what is not allowed for the man to see of a woman like the head and so forth but nothing of the 'awrah (parts considered private).
Based on the above takhsiis of the generality of this Verse has been made. 
Fatima bint Qais reported that Abu 'Amr bin Hafs divorced her absolutely when he was away from home, and he sent his agent to her with some barley. She was displeased with him and when he said:
I swear by Allah that you have no claim on us. she went to Allah's Messenger (sallallaahu alayhi wa sallam) and mentioned that to him. He said: There is no maintenance due to you from him, and he commanded her to spend the 'Idda in the house of Umm Sharik, but then said: That is a woman whom my companions visit. So better spend this period in the house of Ibn Umm Maktum, for he is a blind man and yon can put off your garments. And when the 'Idda is over, inform me. She said: When my period of 'Idda was over, I mentioned to him that Mu'awiya bin Abu Sufyan and Jahm had sent proposal of marriage to me, whereupon Allah's Messenger (sallallaahu alayhi wa sallam) said: As for Abu Jahm, he does not put down his staff from his shoulder, and as for Mu'awiya, he is a poor man having no property; marry Usama bin Zaid. I objected to him, but he again said: Marry Usama; so I married him. Allah blessed there in and I was envied (by others).
al-Nawawi says that in the above mentioned hadith (the hadiith of Aishah) there is evidence that it is permissible for a woman to look at the act of men playing without looking at their bodies. If she looks at their faces with desire than that is haraam by agreement of all scholars.
If she does it without desire and lust and there is no fear of fitnah than there are two views among our Companions and most correct opinion is that it is haraam.
The answer al-Nawawi provides to the hadiith of Aishah is that she looked at their act of playing and not their bodies. The second is that maybe this incident took place before the Verses of not being allowed to look were revealed, and maybe she was under the age of puberty at the time.
Ibn Hajar says in Fath al-Baari:
The zaahir of the tarjamah of the author (meaning Imaam al-Bukhaari) shows that he adopted the opinion of it being permissible for a woman to look at a man who is a stranger (without desire). This is a well known matter and the Shaafi'i school has differed in regards to the raajih opinion. The hadiith of the chapter aids the opinion of those who said it was permitted.
He then goes on to say:
The fact that women were allowed to go the masjid, marketplaces, and on travels while wearing the niqaab so that men would not see them while the men were never told to wear niqaab lest women see them also strengthens the opinion that it is permissible.
And by means of this al-ghazaalii also held the opinion of it being permissible. 
The scholars who permitted it, did so based on the premise that there be no fear of fitnah. We live in fitnah filled times. It is best for women to not look at men at all. It is also the opinion of the majority.
 Tafsir Ibn Kathiir
Volume 6 Page 45
 Tafsir al-Qurtubi
Volume 12 Page 227
 Tafsir al-Qurtubi
Volume 12 Page 228
 Sharh Muslim li Imaam al-Nawawi
Volume 6 Page 184
 Fath al-Baari li Imaam Ibn Hajar al-asqalaani
Volume 9 Page 336-337