The Gesù was conceived in 1551 by Saint Ignatius of Loyola, the founder of the Jesuits Society of Jesus.
When Ignatius died in 1556, the church was still in the drawing stage. But with the determination of Francis Borgia, the third Jesuit general, and patronage from the powerful Farnese family, construction began in 1568, and the building was almost finished by the holy year of 1575.
At its dedication in 1584, it was the largest and first completely new church built in Rome since the sack of 1527.
The Gesù was built with the financial assistance of Cardinal Alessandro Farnese, the richest art patron of his day and the nephew of Paul III, the pope who approved the Society of Jesus. Farnese ended up lavishing 100,000 scudi on the project, an inconceivable fortune at the time, and hired the architects Jacopo Barozzi da Vignola and Giacomo della Porta to design the building.
During the next few decades, the side chapels, transepts, cupola, and high altar area were decorated with precious colored marbles and with paintings by the leading artists of the day.
But suddenly work stopped with Farnese's death in 1589, and the entire ceiling had to wait almost a century before it could be painted, between 1672 and 1685, by the Baroque artist Giovanni Battista Gaulli (Baciccio). It was he who turned the dome into a vision of heavenly splendor and filled the apse with a throng of saints and angels. Baciccio also transformed the ceiling of the nave into a glorification of the name of Jesus, surrounded with gilding and stucco figures who look as if they are diving into the church. Baciccio's paintings were the first major frescoes commissioned in Rome in almost twenty years. His brand of heavenly ceilings were copied throughout Italy and became one of the most characteristic features of Baroque church interiors.
Baciccio's ceilings and dome were not the only major change that took place in the church in the High Baroque era. It was also during these years that the two transepts were dedicated to Ignatius Loyola and Francis Xavier, the first two Jesuits to be canonized, in 1622. Their Relics (St. Ignatius’ remains and St. Francis Xavier’s right hand) are in their respective chapels. For the first time in history the Jesuits could openly celebrate the lives of their own heroes in art and architecture.
The church of the Gesù was conceived as a single hall without aisles to allow for large congregations and provide good acoustics for preaching.
Its plan and decoration, variously described as late Renaissance or early Baroque, would have a tremendous influence on church design in Italy and around the world in subsequent centuries, even inspiring the now-contested term, "the Jesuit style".
(Taken from http://www.chiesadelgesu.org/html/d_storia_chiesa_en.html)