Who Created the Hijri Calendar — and What Exactly Was Invented?

The short answer

No single person invented the Hijri calendar from nothing. The lunar months — Muharram through Dhu al-Hijjah — and the practice of starting each month with a new crescent were already in use in Arabia. What was created was the era: the decision to number years from the Hijra, the migration of the Prophet Muhammad and the early Muslim community from Mecca to Medina. Islamic historical tradition credits that administrative decision to the caliphate of Umar ibn al-Khattab, roughly a generation after the migration itself.

In one line: the months were inherited, the year-numbering was the innovation.

Three things people mean by "the Hijri calendar"

The month cycle. Twelve lunar months, each beginning when the new crescent is seen. The U.S. Naval Observatory describes the Islamic calendar as purely lunar, with months of 29 or 30 days and a year about 11 days shorter than the Gregorian year (USNO, The Islamic Calendar).
The era. A fixed starting point for counting years, written with the marker AH. Without an era, a lunar month cycle tells you the season of the year but not which year you are in.
The tabular (calculated) calendar. An arithmetic version that assigns month lengths by rule instead of by sighting, so that dates can be printed, tabulated, and programmed in advance. The USNO treats this calculated calendar as distinct from the sighting-based one.

Summary: "Who invented it?" usually means "who set the era?" — and that is the part attributed to the early caliphate, not to a single astronomer.

Why the Hijra became the anchor

An era needs an event everyone can agree on and date. The Hijra worked because it marked the formation of a community in Medina — a public, administrative turning point rather than a private one. Years were then counted from the lunar year in which that migration fell, which corresponds to 622 CE. If you want the mechanics of that anchor date, see why the Hijri calendar starts in 622 CE and what AH means after a Hijri year.

What the calendar's framers deliberately did not add

No leap month. Luni-solar calendars insert a thirteenth month to stay aligned with the seasons. The Hijri calendar does not, which is why its dates move earlier through the Gregorian year — about 11 days each year. The USNO's Introduction to Calendars sets out this difference between purely lunar, solar, and luni-solar systems. See also why there is no 13th month.
No fixed year length. A Hijri year runs 354 or 355 days depending on how the months fall.
No single global clerk. Because month starts depend on observation, different countries and committees can announce a new month a day apart.

Summary: the design choice to stay lunar is the reason Ramadan and Hajj rotate through all four seasons over roughly 33 years.

Who maintains it today

For Saudi Arabia's official and administrative dating, the reference is the Umm al-Qura calendar, published by Saudi authorities and used for government documents and date conversion (Umm al-Qura). Pilgrimage scheduling — the dates of Dhu al-Hijjah in particular — is announced through the Saudi Ministry of Hajj and Umrah and the kingdom's official channels rather than by any private calendar publisher.

That split matters when you read a printed calendar: a "civil" Hijri date may be calculated by rule, while Saudi official dates follow the Umm al-Qura reference. The difference is explained in what "civil" means on a Hijri calendar.

Summary: no inventor to name, a traceable era decision, and a modern calendar maintained by national authorities rather than by one individual.