Asshai is a city located on the far side of the explored world from Westeros, in the far south-east of the Essos continent on the Jade Sea. Little is known about it save that it is a great seaport and its people are exotic. Beyond Asshai lies an area known as the Shadow Lands, which are said to lie "under the Shadow". What is casting "the Shadow" is not known, nor is the exact nature of "the Shadow."
Ser Jorah suggests to Daenerys that they flee Khal Drogo's khalasar by night so as not to be caught up in the chaos unleashed when he dies. He says there is a port in Asshai they can head to. Daenerys refuses.
In the books
In the A Song of Ice and Fire novels, Asshai is said to lie on the far south-eastern shores of the Jade Sea, a body of water far to the east even of the Free Cities and Slaver's Bay. It is located close to an enigmatic area called "the Shadow" or "Shadow Lands", and the city is sometimes known as Asshai-by-the-Shadow. The city is usually traveled to by ship, but it can also be reached overland by a circuitous route. The Dothraki consider the city and surrounding area to be ill-omened. The city is reputed to hold large amounts of arcane lore, including knowledge of dragons.
It is not unusual for a voyage from the Free City of Volantis to Asshai and back to take two years.
According to the TV series official pronunciation guide developed for the cast and crew, "Asshai" is pronounced "Uh-SHY", as opposed to "ASS-high", "Uh-SIGH", etc.
The Lands of Ice and Fire places Asshai south-east of Qarth, on the north-eastern shores of the Jade Sea (which has been completely moved from its location on the HBO map). The Saffron Straits link the Jade Sea to as-yet-unknown seas to the east, and Asshai sits at the mouth of the straits. A major river empties into the Jade Sea at Asshai. The river flows out of the Shadow Lands, an area of mountains and hills north-east of the city surrounded by ghost grass. Further upriver from Asshai is the ruined city of Stygai. The large island of Leng lies north-west of Asshai, whilst the fabled Manticore Isles lie south-west of the city. The continent of Ulthos lies just south of Asshai across the Saffron Straits.