LAW & ORDER: CRIMINAL INTENT (2001 - 2011)
This series took the franchise as far from its roots as possible while still being recognizable as a Law & Order show. The original "tweak" this time? Getting to see the criminals do their thing in tandem with the cops trying to find them. Pretty quickly, though, it became a star-vehicle for the crime-solving chemistry between Vincent D'Onofrio's Detective Goren (a brooding teddy-bear of a man, possibly suffering mild autism but gifted with a socially-crippling genius for getting into the heads of criminal masterminds) and Katherine Erbe's Eames, his more traditionally-focused partner.
Instead, shared-focus with criminals became a mandate to hand Goren a succession of supervillains to match wits with master-criminals with elaborate plans and grandiose justifications, most notably Olivia D'Abo as brilliant, sexy serial-killer Nicole Wallace, who ultimately became Goren's arch-nemesis for much of the series. For awhile (to accommodate health and personal issues of D'Onofrio's) half the episodes got handed to Chris Noth as Det. Mike Logan, a fan-favorite returning character from the original Law & Order's golden years; followed by a season-length digression with a new Goren-esque character played by (of all people) Jeff Goldblum.
LAW & ORDER: TRIAL BY JURY (2005 - 2006)
Whereas the previous two spin-offs had kicked the lawyer side of the original series' double-act to the sidelines, TBJ goes all-law, all the time, with each episode covering a court case starting with both side's pre-trial preparations. Interesting premise, but "straight" lawyer shows (as opposed to ensemble dramadies whose characters were lawyers) haven't been able to gain traction in years. The series lasted only 13 episodes.
LAW & ORDER: UK (2009 -2014)
A novel premise - porting the L&O tone and trappings to the legal system of another country - yielded this respectably popular eight series (53 episodes total) ITV series, which follows the original series model of splitting storylines between detectives and Crown Prosecutors. Went on official hiatus last year, but has been tipped for a potential comeback since.
LAW & ORDER: LA (2010 - 2011)
This sounded like a can't-miss idea: End the original Law & Order, by this point eclipsed in popularity by the CSI/NCIS era of procedurals and by the megapopularity of it's own SVU sibling, and revive the concept in Los Angeles. It had a good cast, too, including Skeet Ulrich, Alfred Molina and Terence Howard. But it just couldn't stick the landing (even with Ulrich's character getting a crossover-cameo with SVU in which Det. Benson practically knights him as the new coolest-cop-on-Earth for the audience) and vanished after only a season.