He was the son of Edmund Fitzalan, 9th Earl of Arundel and Alice Warenne. His birthdate is uncertain, but was not before 1307.
Around 1321 Richard's father allied with king Edward II's favorites the Despensers, and Richard was married to Isabella, daughter of the younger Hugh le Despenser. Fortune turned against the Despenser party, and in 1326 Richard's father was executed, and Richard did not succeed to his father's estates or titles.
However by 1330 political conditions had changed, and over the next few years Richard was gradually able to reaquire the earldom of Arundel as well as the great estates his father had held in Sussex and in the Welsh Marches. Beyond this, in 1334 he was made justice of North Wales (later his term in this office was made for life), sheriff for life of Carnarvonshire, and governor of Carnarvon Castle.
Despite his high offices in Wales, in the following decades Arundel spent much of his time fighting in Scotland and France. In 1337 he was joint commander of the English army in the north, and the next year he was made the sole commander.
In 1340 he fought at the Battle of Sluys, and then at the siege of Tournai. After a short term as warden of the Scottish Marches, he returned to the continent, where he fought in a number of campaigns, and was appointed joint Lieutanant of Aquitaine in 1340.
Arundel was one of the three principal English commanders at the Battle of Crecy. He spent much of the following years on various military campaigns and diplomatic missions.
In 1353 he succeeded to the Earldom of Warenne (or Surrey), which even futher increased his great wealth. He made very large loans to Edward III but even so on his death left behind a great sum in hard cash.
Arundel married twice. His first wife (as mentioned above), was Isabella Despenser. He repudiated her, and had the marriage annulled on the grounds that he had never freely consented to it. After the annullment he married Eleanor Plantagenet, daughter of Henry Earl of Lancaster.
By his first marriage he had one daughter. By the second he had 3 sons: Richard, who succeeded him as earl; Thomas, who was a Marshall of England, and drowned in 1379; and Thomas Arundel, who became Archbishop of Canterbury. He also had 2 surviving daughters by his second wife: Joan, who married Humphrey de Bohun, earl of Hereford, and Alice, who married Thomas Holland, 2nd Earl of Kent.