January 31 – Henry, King of Portugal dies with no direct heirs, precipitating a Succession crisis.
March 1 – Michel de Montaigne signs the preface to his most significant work, Essays. They are published later the same year.
March 25 – Iberian Union: King Philip II of Spain becomes King of Portugal under the name Philip I, following the death without heirs of King Henry of Portugal, in a personal union of the crowns, thus maintaining Portuguese independence (in Europe and throughout the Portuguese Empire). The Philippine Dynasty rule lasts until 1640.
April 6 – Dover Straits earthquake.
June – England signs a commercial treaty with the Ottoman Empire.
June 11 – Juan de Garay founds Buenos Aires.
June 25 – The Book of Concord, a collection of Lutheran confessional documents, is published.
July 12 – The Ostrog Bible, the first complete printed Bible translation into a Slavic language (Old Church Slavonic), is first printed at Ostroh in the Polish–Lithuanian Commonwealth (modern-day Ukraine) by Ivan Fyodorov.
August 25 – Battle of Alcântara: Spanish armies, led by Fernando Álvarez de Toledo, defending the claim of King Philip II of Spain to the Portuguese throne, defeat the armies of Portuguese claimant António, Prior of Crato.
September 26 – Francis Drake returns to Plymouth in England from his voyage of circumnavigation (westabout) on the Golden Hind, the second completed in a continuous voyage and the first under its original commander.
The Billy Mitchell volcano on the island of Bougainville undergoes a catastrophic eruption (VEI 6).
The first session of the Jewish Vaad (Council of Four Lands) is held in Lublin, Poland; 70 delegates of Jewish local qahals meet to discuss taxation and other issues important to Jewish communities.
Old City of Zamość established in Poland by Jan Zamoyski.
Jesuit missionaries arrive at the court of Akbar, Ruler of the Mughal Empire.
January 8 – Jens Hermansson Juel, Danish governor of Norway (d. 1634)
Jan Baptist van Helmont, Flemish chemist (d. 1644)
Alexander Ruthven, Scottish earl (d. 1600)
January 20 – Stefano Amadei, Italian painter (d. 1644)
January 29 – Willem Isaacsz Swanenburg, Dutch engraver (d. 1612)
January 30 – Gundakar, Prince of Liechtenstein, court official in Vienna (d. 1658)
c. January – John Smith, English explorer and Virginia settler (d. 1631)
February – John Digby, 1st Earl of Bristol, English diplomat (d. 1653)
February 1 – Francis Fane, 1st Earl of Westmorland, English noble (d. 1629)
February 2 – Jens Bjelke, Norwegian noble (d. 1659)
February 22 – Charles de l'Aubespine, marquis de Châteauneuf, French diplomat and government official (d. 1653)
February 24 – Matthias Hoe von Hoenegg, German theologian (d. 1645)
Orazio Giustiniani, Catholic cardinal (d. 1649)
Giovanni Srofenaur, Italian musician (d. 1634)
March 31 – Bogislaw XIV, Duke of Pomerania (d. 1637)
Augusta of Denmark, Duchess Consort of Holstein-Gottorp (1596-1616) (d. 1639)
William Herbert, 3rd Earl of Pembroke, English noble, courtier and patron of the arts (d. 1630)
April 15 – George Calvert, 1st Baron Baltimore, English politician and colonizer (d. 1623)
April 18 – (baptism) Thomas Middleton, English playwright (d. 1627)
Miguel Avellán, Roman Catholic prelate, Auxiliary Bishop of Toledo from 1633 (d. 1650)
Vincent de Paul, French priest and saint (d. 1660)
Johann Faulhaber, German mathematician (d. 1635)
Richard Webb, English settler in America (d. 1665)
May 6 – Charles Gonzaga, Duke of Mantua and Montferrat, French noble (d. 1637)
May 14 – Bassam Al-Soukaria, Lebanese army commander (d. 1667)
May 30 – Fadrique de Toledo, 1st Marquis of Villanueva de Valdueza, Spanish noble and admiral (d. 1634)
June 6 – Godefroy Wendelin, Flemish astronomer (d. 1667)
June 9 – Daniel Heinsius, Dutch scholar (d. 1655)
June 12 – Adriaan van Stalbemt, Flemish Baroque painter (d. 1662)
June 14 – Elisabeth Magdalena of Pomerania, German duchess (d. 1649)
June 26 – Gaspar de Borja y Velasco, Spanish cardinal (d. 1645)
July 5 – Carlo Contarini, Doge of Venice (d. 1656)
July 6 – Johann Stobäus, German composer (d. 1646)
July 10 – Humphrey Chetham, English merchant (d. 1653)
July 18 – Giovanni Giacomo Semenza, Italian painter (d. 1638)
July 29 – Francesco Mochi, Italian early-Baroque sculptor (d. 1654)
August 2 – Prince Jeongwon, Korean prince (d. 1619)
August 19 – Pierre Vernier, French mathematician (d. 1637)
August 24 – John Taylor, English poet (d. 1654)
September 4 – George Percy, English explorer (d. 1632)
Francisco de Quevedo, Spanish writer (d. 1645)
Robert Gordon of Straloch, Scottish cartographer (d. 1661)
Charles Annibal Fabrot, French lawyer (d. 1659)
Thomas Fanshawe, English politician (d. 1631)
September 17 – Countess Charlotte Brabantina of Nassau, Belgian noble (d. 1631)
September 24 – Elisabeth of Schleswig-Holstein-Sonderburg, Duchess consort of Pomerania (d. 1653)
October 8 – Gábor Esterházy (1580–1626), Hungarian noble (d. 1626)
October 12 – Hortensio Félix Paravicino, Spanish preacher and poet from the noble house of Pallavicini (d. 1633)
October 20 – Peter Crüger, German astronomer and mathematician (d. 1639)
October 30 – Armand-Nompar de Caumont, duc de La Force, Marshal of France (d. 1675)
November 9 – Johannes Narssius, Dutch physician and poet (d. 1637)
December 1 – Nicolas-Claude Fabri de Peiresc, French astronomer (d. 1637)
Samuel Argall, English adventurer and naval officer (d. 1626)
Nabeshima Katsushige, Japanese daimyō (d. 1657)
Philipp Clüver, German geographer and historian (d. 1623)
Dirk Hartog, Dutch ship's captain and explorer (d. 1621)
Jean Jannon, Swiss-born typefounder (d. 1658)
Robert Killigrew, English courtier, politician, ambassador and knight (d. 1633)
Willebrord Snellius, Dutch astronomer and mathematician (d. 1626)
Raphael Sobiehrd-Mnishovsky, Bohemian lawyer and writer (d. 1644)
Pierre Vernier, French mathematician and instrument inventor (d. 1637)
Krzysztof Zbaraski, Polish nobleman (d. 1627)
William Brabazon, 1st Earl of Meath, English noble (d. 1651)
Edward Fairfax, English translator (d. 1635)
Frans Hals, Dutch painter (d. 1666)
Zacharias Janssen, Dutch spectacle-maker (d. 1638)
Alexander Leslie, 1st Earl of Leven, Scottish soldier (d. 1661)
Benjamin, Duke of Soubise, French Huguenot leader (d. 1642)
Adriana Basile, Italian composer (d. 1640)
January 18 – Antonio Scandello, Italian composer (b. 1517)
January 31 – King Henry of Portugal (b. 1512)
February 2 – Bessho Nagaharu, Japanese retainer (b. 1558)
February 24 – Henry FitzAlan, 19th Earl of Arundel, English nobleman (b. 1511)
February 28 – Krystyna Radziwiłł, Polish noblewoman (b. 1560)
May 3 – Thomas Tusser, English poet and farmer (b. c. 1524)
June 10 – Luís de Camões, Portuguese poet (b. c. 1524)
July 9 – Zofia Odrowąż, Polish noblewoman (b. 1540)
August 1 – Albrecht Giese, German politician and diplomat (b. 1524)
August 19 – Andrea Palladio, Italian architect (b. 1508)
August 20 – Jerónimo Osório, Portuguese historian (b. 1506)
August 30 – Emmanuel Philibert, Duke of Savoy (b. 1528)
October 5 – Matsudaira Shigeyoshi, Japanese general (b. 1493)
November 3 – Jerónimo Zurita y Castro, Spanish historian (b. 1512)
November 30 – Richard Farrant, English composer (b. 1530)
December 1 – Giovanni Morone, Italian cardinal (b. 1509)
Giovanni Filippo Ingrassia, Italian anatomist (b. 1545)
Ruy López de Segura, Spanish priest and writer on chess (b. 1530)
Inés de Suárez, Spanish conquistadora (b. 1507)
John Heywood, English dramatist (b. 1497)
Robert Lindsay of Pitscottie, Scottish chronicler (b. c. 1532)
Year 1580 (MDLXXX) was a leap year starting on Friday (link will display the full calendar) of the Julian calendar, and a leap year starting on Tuesday (link will display the full calendar) of the Proleptic Gregorian calendar.