I have a question regarding taking photos and making videos of living creatures. I want to know is this lawful according to Qur'an and Sunnah to take photos and make videos of one's family and keep them as a memory and enjoy watching good events captured by these means.

Praise be to Allah, the Lord of the Worlds; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Forming images are of three kinds:
1. Forming three-dimensional figures of living creatures by chiseling, molding or carving from solid material. Such images are forbidden in Shariah. There is no disagreement about that for the Hadith of the Prophet: "Image makers will be most severely punished by Allah on the Day of Resurrection". [Reported by Imams Bukhari and Muslim]
Imam Tirmizi reported on the authority of Ibn Abbas that the Prophet said: "A person who eavesdrops upon a party will have molten lead poured into his ears on the Day Judgement, and one who paints the portrait of a living being will be punished and called upon to infuse life into it which he will not be able to do".
2. Making pictures by hand such as paintings on walls, prints on clothes drawing on papers, etc. The majority of the Muslim scholars are of the opinion that such images are also prohibited in Shariah. They generalize the rule of prohibiting the imitation of Allah in Creating to these pictures also. The Hadith on which they base this rule is that the Prophet said: "The most grievous torment for the people on the Day of Resurrection would be for those who try to imitate Allah in the act of Creation". [Reported by Imam Bukhari]
3. Photography by camera or video. The Muslim scholars have two different opinions concerning this matter. Some of them permit it while others do not permit it. We side with those who say that if the photography does not contain any forbidden matter such as the pictures of uncovered women or the pictures made for the purpose of honouring certain Creatures, etc. then there is no harm in such photography. Since the reason for which the photography of living beings is forbidden, i.e. imitating of Allah's Creation, does not exist in photography of camera or video. But it is merely taking a photo of what Allah has Created, especially if there is a legal purpose such as taking a passport photo, making photo to show the performance of Mujahideen or recording Islamic lectures, etc. then it is permitted.
In spite of the details mentioned above we emphasize the fact that avoiding all kinds of photography is better to avoid doubts and to the differing opinions of scholars.
As for hanging the pictures of living beings, it is safer not to hang them.
However, making images, prints, statues and photographs of inanimate objects, such as trees, moon, sun and mountains, is absolutely permissible.
Allah knows best.