Yes, cats are allowed in Islam.
It is permissible to keep cats in the house, and there is nothing wrong with that because cats are not harmful or naajis (impure).
With regard to cats not being harmful, no one disputes that rather they are useful because they eat snakes, rats bugs and other things that may be in the house or yard.
With regard to cats not being naajis, this is known from the hadeeth of Kabshah bint Ka’b ibn Maalik, who said that Abu Qutaadah — her husband’s father — entered upon her and she poured water for him to do wudoo’, and a cat came to drink from it, so he tipped the vessel for it to drink. Kabshah said:
“He saw me looking at him and said, “Do you find it strange, O daughter of my brother?’ I said, ‘Yes.’ He said, ‘The Messenger of Allah (peace and blessings of Allah be upon him) said, “They (cats) are not naajis, rather they are of those who go around amongst you.”-(Narrated by al-Tirmidhi, 92).
Note: It is permissible to keep cats but it is not permissible to buy or sell them; they may be given as gifts or given away. That is because of the hadeeth of Abu’l-Zubayr who said: “I asked Jaabir about the price of dogs and cats. He said:
‘The Prophet (peace and blessings of Allah be upon him) forbade that.”-(Narrated by Muslim, 1569).