Answer by taking a quote from the following article:

Jarmusch was born January 22, 1953, in Cuyahoga Falls, Ohio, the middle of three children of middle-class suburbanites. His mother, of German and Irish descent, had been a reviewer of film and theatre for the Akron Beacon Journal before marrying his father, a businessman of Czech and German descent who worked for the B.F. Goodrich Company. She introduced Jarmusch to the world of cinema by leaving him at a local cinema to watch matinee double features such as Attack of the Crab Monsters and Creature From the Black Lagoon while she ran errands. The first adult film he recalls seeing was the 1958 cult classic Thunder Road, the violence and darkness of which left an impression on the seven-year-old Jarmusch.

In 1986, Jarmusch wrote and directed Down by Law, starring musicians John Lurie and Tom Waits, and Italian comic actor Roberto Benigni (his introduction to American audiences) as three convicts who escape from a New Orleans jailhouse. Shot like the director's previous efforts in black and white, this constructivist neo-noir was Jarmusch's first collaboration with Dutch cinematographer Robby Muller, who had been known for his work with Wenders.  His next two films each experimented with parallel narratives: Mystery Train (1989) told three successive stories set on the same night in and around a small Memphis hotel, and Night on Earth (1991) involved five cab drivers and their passengers on rides in five different world cities, beginning at sundown in Los Angeles and ending at sunrise in Helsinki. Less bleak and somber than Jarmusch's earlier work, Mystery Train nevertheless retained the director's askance conception of America. He wrote Night on Earth in about a week, out of frustration at the collapse of the production of another film he had written and the desire to visit and collaborate with friends such as Benigni, Gena Rowlands, Winona Ryder and Isaach de Bankole.  As a result of his early work, Jarmusch became an influential representative of the trend of the American road movie. Not intended to appeal to mainstream filmgoers, these early Jarmusch films were embraced by art house audiences, gaining a small but dedicated American following and cult status in Europe and Japan. Each of the four films had its premiere at the New York Film Festival, while Mystery Train was in competition at the 1989 Cannes Film Festival. Jarmusch's distinctive aesthetic and auteur status fomented a critical backlash at the close of this early period, however; though reviewers praised the charm and adroitness of Mystery Train and Night On Earth, the director was increasingly charged with repetitiveness and risk-aversion.  A film appearance in 1989 as a used car dealer in the cult comedy Leningrad Cowboys Go America further solidified his interest and participation in the road movie genre. In 1991 Jarmusch appeared as himself in Episode One of John Lurie's cult television series Fishing With John.

What was Down by Law about?
In 1986, Jarmusch wrote and directed Down by Law, starring musicians John Lurie and Tom Waits, and Italian comic actor Roberto Benigni