Legislation making the changes – which was passed by MPs at Westminster – came into force at midnight.
The first same-sex weddings in the North of Ireland are set to take place on Valentine’s Day, February 14, 2020.
The government has until the end of March to come up with regulations for the provision of abortion services.
The legislation took effect after the 21 October deadline passed without a devolved government being re-formed.
The DUP and some politicians from other unionist parties triggered a recall of the Assembly in an attempt to block the lifting of a near-ban on terminations.
The assembly at Stormont had not sat for more than 1,000 days after devolved government collapsed when power-sharing coalition partners the DUP and Sinn Féin split in a bitter row.
But the move failed because a new speaker could not be elected on a cross-community basis.
SDLP leader Colum Eastwood led his party out of the chamber branding it as a “political sham”.
Before now, abortion was only allowed in Northern Ireland if a woman’s life was at risk or there was a danger of permanent and serious damage to her physical or mental health.
Section 58 and Section 59 of the Offences Against the Person Act 1861 – which made abortion a criminal offence – have been repealed.
The Northern Ireland (Executive Formation etc) Act 2019 has also placed a duty on the government to implement the recommendations of a report by the United Nations Committee on the Elimination of Discrimination Against Women (CEDAW), published in 2018.
The CEDAW report said abortion should be legalised where there is a threat to a pregnant woman’s mental or physical health, without the conditionality of “long-term or permanent” effects.
It recommended terminations should be permitted in cases of rape or incest.
The committee also said abortions should be allowed where there is “severe fetal impairment”, but that provision should not “perpetuate stereotypes” towards disabled people.
It added that social and financial support should be ensured for women who decided to carry such pregnancies to term.A further series of recommendations included providing access to “high quality abortion and post-abortion care in all public health facilities”, and making “age-appropriate, comprehensive and scientifically accurate education” on “sexual and reproductive health and rights” a compulsory part of the curriculum.
The government in London will decide on more detailed measures to fulfill the requirements of the legislation.
British Secretary of State Julian Smith is mandated to put in place regulations by 31 March 2020.
The British government has issued guidance to medical professionals which covers the period from now until that date.
Same-sex marriages have been allowed in England, Scotland and Wales since 2014, but Stormont did not legalise them.
The last of five votes on the issue in the devolved assembly – in November 2015 – resulted in a numerical majority in favour of same-sex marriage for the first time.
But the DUP blocked a change in the law by using a veto known as the Petition of Concern.
The new legislation says the Westminster government must bring in regulations to provide for same-sex marriage by 13 January 2020.
Because couples have to indicate their intention to marry 28 days before doing so – the first gay weddings are expected to be held in the week of Valentine’s Day.
The North’s Catholic bishops said Monday was a tragic day for unborn children and a sad day for local democracy.
They say they are also concerned at the redefinition of marriage and appealed to the political parties to re-double their efforts to restore the power-sharing executive.