The 76-year-old pontiff officially began his ministry on Tuesday morning, in front of hundreds of thousands of people at St Peter's Square.
Among the crowd were 132 world political and religious leaders, including six sovereign rulers, 31 heads of state and three princes.
They included Irish President Michael D Higgins, who Pope Francis spoke to after the ceremony during a brief encounter inside the Basilica.
"Best wishes to the people of Ireland," the pontiff said. "And I ask them for their prayers."
Speaking in Spanish, Mr Higgins, replied: "On behalf of the Irish people, who hold a special affection for the people of South America, I want to wish you every success in your vision for the Church.
"My wish for you on behalf of the people of Ireland is that you have the best of health in your mission on behalf of all humanity."
Cardinal Jorge Mario Bergoglio became the first Latin American pope when he was elected last week by the conclave to represent the world's 1.2bn-strong Catholic Church.
The Argentinian received the symbolic ring and the pallium, a wool stole, to mark the beginning of his papacy at the inauguration mass.
During his inaugural sermon, Pope Francis called on world leaders to respect the "beauty of the created world" and protect the weakest in society.
He said: "Please, I would like to ask all those who have positions of responsibility in economic, political and social life, and all men and women of goodwill - let us be 'protectors' of creation, protectors of God's plan inscribed in nature, protectors of one another and of the environment."
As the service got underway, Francis toured St Peter's Square where he kissed babies and blessed a disabled man.
The crowds applauded as the 266th pope paid tribute to his "venerable predecessor", Benedict XVI, and he also quoted from Pope John Paul II.