Nasta`līq is one of the main genres of Islamic calligraphy. It was developed in Iran in the 14th and 15th centuries. Although it is sometimes used to write Arabic text, and is quite frequently used for titles and headings, it has been more popular in the Persian, Turkic, and South Asian spheres of influence. Nastaʿlīq has extensively been (and still is) practiced in Iran and Afghanistan as a form of art. A less elaborate version of Nastaʿlīq serves as the preferred style for writing Persian, Pashto and Urdu. The Nastaʿlīq script was also used for writing Ottoman Turkish, where it is known as ta`liq (not to be confused with a totally different Persian style, also called ta`liq).
Nastaʿlīq is amongst the most fluid calligraphy styles for the Arabic alphabet. It has short verticals with no serifs, and long horizontal strokes. It is written using a piece of trimmed reed with a tip of 5–10 mm, called "qalam" ("pen," in Arabic), and carbon ink, named "davat." The nib of a qalam is usually split in the middle to facilitate ink absorption.
After the Islamic conquest of Persia, Iranians adopted the Perso-Arabic script and the art of Arabic calligraphy flourished in Iran alongside other Islamic countries. Apparently, Mir Ali Tabrizi (14th century) developed Nastaʿlīq by combining two existing scripts of Nasḫ and Taʿlīq. Hence, it was originally called Nasḫ-Taʿlīq.
The Mughal Empire used Persian as the court language during their rule over the Indian subcontinent. During this time, Nastaʿlīq came into widespread use in South Asia, including Pakistan, India, and Bangladesh. The influence remains to this day. In Pakistan, almost everything in Urdu is written in the script, concentrating the greater part of Nasta’līq usage in the world. In Hyderābād, Lakhnau, and other cities in India with large Urdu-speaking populations, many street signs and such are written in Nastaʿlīq. The situation of Nastaʿlīq in Bangladesh used to be the same as in Pakistan until 1971, when Urdu ceased to remain an official language of the country. Today, only a few neighborhoods (mostly inhabited by Bihāris) in Dhaka and Chittagong retain the influence of the Persian and Nastaʿlīq.

An example of the Nastaʿlīq script used for writing Urdu
The first publicly available attempt at developing a Unicode-based OpenType Nastaleeq font was Nafees Nastaleeq. This font was developed by FAST University in Pakistan by a team of four people led by Dr. Sarmad Hussain, others including Aamir Wali, Aatif Gulzar and calligrapher Mr. Jameel-ur-Rehmaan. This team spent 18 months to develop Nafees Nastaleeq following the Lahori Ravish of Nastaleeq. It has 900+ shapes, 103 joining rules, 77 mark placement rules, 15 kerning rules, 24 cursive attachments and 30+ ligatures. Nafees Nastaleeq was to be open source as it was funded by some American grant, but later on the team decided not to disclose its internals, and as a result the font is still free to use but sources are not available to the public. Due to massive joining and mark placement rules, this font has serious performance issues. Nafees Nastaleeq makes the rendering process quite slow on larger amounts of text.
Later on, Dr. Attash Durrani of the Center of Excellence for Urdu Informatics initiated a project to develop a standard Unicode-based OpenType Nastaleeq font named Pak Nastaleeq funded by the government of Pakistan. Mohsin Shafique Hijjazi was responsible for the implementation and contextual analysis of Nastaleeq as they were not publicly available at that time. Using some mathematical modeling, he reduced the joining rules first from 100 to 25 and then from 25 to only two. This font is still in its beta stages and resembles Noori Nastaleeq following Dehelvi Script. Only 200 shapes, two joining rules, five mark placement rules, one cursive attachment rule, and no ligatures, this font is highly efficient to use and is targeted for both desktop publishing and the World Wide Web.
In Iran, for a long time, yet the only software able to develop Nastaleeq Script was named Zarnegar. I never tried it because of its expensiveness. But recently one of my friends sent the Unicode based Nastaleeq font, which contains Persian, Urdu, Latin, Greek, Cyrillic, Hebrew and Thai characters.
I made a tableau using this font for you, in the Pdf format (355 Kb). It's a poem of mine, Hope to enjoy it!
See also:
A Tableau in the Pahlavic Language
An Album of Persian Calligraphies