Varadkar, 38, who was earlier this month elected leader of the ruling Fine Gael party, is a medical doctor and the country's youngest and first gay leader.
A parliamentary vote today confirmed Varadkar's position as Ireland's new taoiseach (prime minister) with 57 votes in favour, 50 votes against, and 47 abstentions, according to BBC News. "The government I lead will not be one of left or right, it will be one of a new European center where everyone gets a fair go".
He also became the first openly gay man elected to lead the Republic of Ireland, where homosexuality was illegal until just 24 years ago.
"Leo was definitely laying the ground for disappointment and for the most part people seemed to accept that", said one source who was at the meeting.
Leo Varadkar visited Galway a year ago, spending a day-and-a-half checking out state-sponsored jobs programs in the west of Ireland before heading to Eyre Square for a late-night snack.
Last month, Kenny resigned as leader of Fine Gael under pressure from both inside and outside of the party.
He then travelled to the president's residence, Aras an Uachtarain, where President Michael D Higgins gave him the seals of office to officially confirm his appointment. While Trinity has seen plenty of its graduates enjoy successful political careers, including current Chancellor of the University of Dublin, Mary Robinson, who is a former President of Ireland, it has never had a graduate elected taoiseach before.
It says: "Although regarded as relatively liberal on social issues such as gender equality and abortion rights, Varadkar has been criticised by opposition parties for his right-wing economic views".
Mr Varadkar was appointed with the support of 58 TDs including Fine Gael, the Independent Alliance, two Independent Ministers and Independent TD Michael Lowry.
Leo Varadkar was first elected as a councillor at the age of 24 and was elected to the Irish Parliament in 2007.
Minister for Transport, Tourism and Sport Shane Ross is also believed to have met with Varadkar.