THERESA May and Jeremy Corbyn locked horns during Prime Minister's Questions, trashing each other's record in dealing with allegations of racism within their parties.

The PM switched focus on to Labour's approach to anti-Semitism after Mr Corbyn began by asking about the UK Government's record on climate change.

The Leader of the Opposition pushed back with criticism of the Conservative Party's response to Islamophobia and the views of some of its members.

On climate change, he also claimed that at the current rate the government would not meet its 2050 net-zero emissions target "until 2099"; warning at this point it would be "too late for our planet and our children".

READ MORE: New PM's war chest to crush indyref2 hopes 

Mr Corbyn started by asking why the Government had been accused of "coasting" over climate change, to which Mrs May replied: "The Government has a fine record on climate change, including our recent legislation on net zero emissions.

"But there is an issue that needs to be addressed in this House, and before (Mr Corbyn) stands up and parades himself as the champion of climate change or the champion of the people or the defender of equality and fairness, he needs to apologise for his failure to deal with racism in the Labour Party."

The PM held up a newspaper advertisement, telling MPs: "Just today, 60 distinguished members of the Labour Party have written in the newspapers 'the Labour Party welcomes everyone ... except, it seems, Jews. This is your legacy Mr Corbyn. You still haven't opened your eyes. You still haven't told the whole truth. You still haven't accepted your responsibility. You have failed the test of leadership'. Apologise now."

Mr Corbyn hit back, saying: "This party was the first to introduce anti-racist legislation into law in Britain. This party totally opposes racism in any form whatsoever.

"Anti-Semitism has no place in our society, no place in any of our parties, and no place in any of our dialogue. Neither does any other form of racism.

"And when 60 per cent of Tory Party members think Islam is a threat to Western civilisation, and the Prime Minister has said she will act on Islamophobia within her own party, I hope she does.

"I look forward to that being dealt with as we deal with any racism that occurs within our own party as well."

Mr Corbyn switched back to climate change before Mrs May said: "I note[he] did not apologise in response to my first question.

READ MORE: Jeremy Hunt and Boris Johnson trade blows over Brexit, leadership and Trump in live head to head 

"We deal with Islamophobia in the Conservative Party, any allegations of Islamophobia are dealt with; unlike his way in the Labour Party where he is failing to deal with anti-Semitism.

"And he can stand up and say all he likes about the Labour Party introducing anti-racism legislation.

"Just last week, Trevor Phillips, the former Chairman of the Equality and Human Rights Commission said the following: 'Labour today presents like a textbook case of institutional racism'."

Mr Corbyn repeated his party's opposition to racism, declaring: "Coming from a Prime Minister who encouraged the hostile environment, sent 'go home' vans around London and deported British citizens - which she's now had to compensate them for - I think she might look to her own party and own Government's records as well."

After Mr Corbyn raised further issues on climate change, Mrs May said: "Still no apology, I note."

In his final remarks, Mr Corbyn said: "Air pollution levels breached legal limits in 37 of the 43 areas of this country.

"Two-thirds of our children are growing up in an area where pollution breaches legal limits. This crisis is literally suffocating our children and damaging their health.

"Once again, this Government is dodging its responsibility, while Labour leads the way."

Mrs May responded by highlighting how tens of thousands of new jobs linked to renewables and clean growth before warning she would not take "any lectures" from Labour on the issue.

"He talks about dodging responsibility; the person who has been dodging his responsibility during this PMQs is the right honourable gentleman.

"The real disgrace is his handling of racism in the Labour Party. Activists protesting, MPs leaving, staff resigning. What would his great heroes of Attlee, Bevan and Benn think?

"Look what he has done to their party, we will never let him do it to our country," she added to roars of "more" from Tory backbenchers.