January 4 – The Fabian Society is founded in London.
January 5 – Gilbert and Sullivan's Princess Ida premières at the Savoy Theatre, London.
January 18 – William Price attempts to cremate his dead baby son, Iesu Grist, in Wales. Later tried and acquitted on the grounds that cremation is not contrary to English law, he is thus able to carry out the ceremony (the first in the United Kingdom in modern times) on March 14, setting a legal precedent.
February 1 – A New English Dictionary on historical principles, part 1 (edited by James A. H. Murray), the first fascicle of what will become The Oxford English Dictionary, is published in England.
February 5 – Derby County Football Club is founded in England.
March 13 – The siege of Khartoum, Sudan, begins (ends on January 26, 1885).
March 28 – Prince Leopold, the youngest son and the eighth child of Queen Victoria and Prince Albert, dies, aged 30 in Cannes, France.
March – John Joseph Montgomery conducts the first manned glider flights in the United States near Otay, California.
April 20 – Pope Leo XIII publishes the encyclical Humanum genus, denouncing Freemasonry and certain liberal beliefs which he considers to be associated with it.
April 22
A German protectorate is established over South-West Africa.
The Colchester earthquake, England, the UK's most destructive, occurs.
May 1 – The eight-hour workday is first proclaimed by the Federation of Organized Trades and Labor Unions in the United States. This date, called May Day or Labour Day, becomes a holiday recognized in almost every industrialized country.
May 16
Angelo Moriondo of Turin is granted a patent for an espresso machine.
Sweden's Finance Minister Robert Themptander becomes his country's Prime Minister (1884–88).
June 4 (N.S.) (May 23 O.S.) – The future flag of Estonia is consecrated as the flag of the Estonian Students' Society.
June 13 – LaMarcus Adna Thompson opens the "Gravity Pleasure Switchback Railway" at Coney Island, New York City.
June 28 – The Norwegian Association for Women's Rights is founded.