Romer attended Trinity Hall at Cambridge University and was Senior Wrangler in 1863, and also won the Smith's Prize in that year. From 1865 to 1866, he was a professor of mathematics at Queen's College, Cork, but was called to the bar in 1867. He became QC in 1881, and a bencher of Lincoln's Inn in 1884.
Romer was sworn a Privy Councillor in 1899, and elected a Fellow of the Royal Society in the same year. He was appointed Knight Grand Cross of the Order of the Bath (GCB) in the New Year Honours list 1 January 1901, and was invested by King Edward VII in February the same year.
In 1864 he married Betty Lemon, daughter of Mark Lemon, editor of Punch. Their son was Mark Romer, Baron Romer and their grandson was Sir Charles Robert Romer; both were also Lords Justice of Appeal. Their daughter Helen Mary married future Lord Chancellor Frederic Maugham.
- GCB : Knight Grand Cross of the Most Honourable Order of the Bath - 1 January 1901 - New Year Honours list
- Venn, J.; Venn, J. A., eds. (1922–1958). "Romer, Robert". Alumni Cantabrigienses (10 vols) (online ed.). Cambridge University Press.
- MacKinnon, Frank Douglas; (rev.) Patrick Polden (2004). "Romer, Sir Robert (1840–1918)". Oxford Dictionary of National Biography. Oxford University Press. Retrieved 19 February 2011.