The visit is their first trip overseas as a married couple, following their wedding in May.
It comes "at the request of Her Majesty's Government", a spokesperson for Kensington Palace said.
Later, they will take part in a Gaelic sports festival where traditional Irish sports like hurling, camogie and rounders will be played.
The Book of Kells is one of Ireland's greatest cultural treasures, and is on display at Trinity College in the heart of Dublin.More news: Froome faces challenge from BMC team at Tour de France
Other visits include DogPatch Labs to meet youth learning coding through free workshops.
Meghan and Harry arrived at the British Ambassador's residence in Glencairn House holding hands and looking as loved up as ever.
Even before leaving for Ireland, it has been a busy week for the newlyweds.
The American actress turned British royal, who was just yesterday criticized by social media fashionistas for wearing a "drab" olive green ensemble to Prince Louis' christening, set the world ablaze in a stunning custom Dior hours later when she joined fellow royals to celebrate Royal Air Force's 100th anniversary celebrations at Westminster Abbey this morning.More news: Mahrez signs five-year deal with Man City
The Duke and Duchess of Sussex will visit Dublin, Ireland on Tuesday 10th and Wednesday 11th July.
However neither Prince Harry nor Meghan were named among Louis' six godparents, with the honour going to Guy Pelly, who was once dubbed the "court jester", family friend Nicholas van Cutsem and William's school pal Harry Aubrey-Fletcher.
William and Kate's youngest is now fifth in line to the throne, with older siblings Prince George and Princess Charlotte fourth and fifth respectively.
Keeping the rest of her look very simple, Meghan added diamond and gold Cartier stud earrings, which were on show with her brunette hair pulled back in a low bun.More news: The most interesting late development in Trump's Supreme Court derby
More than 2,200 people packed out Westminster Abbey for the service, with dignitaries including Defence Secretary Gavin Williamson and the Archbishop of Canterbury Justin Welby attending.