a.k.a Inverted Coaster, SLC, Leg's Free Coaster...

B&M Invert

B&M first thought of making this coaster when riding a ski-lift in Switzerland.  Soon Batman: The Ride opened in Six Flags Great America, which sparked countless Batman: The Ride clones built throughout the USA.  Whilst Batman: The Ride was under construction, a young John Wardley managed to get B&M to agree to build one at Alton Towers - to be Alton's first major roller coaster (Nemesis). Cedar Fair also began negotiations with B&M to build the world's tallest and fastest Inverted coaster of that time (Raptor).  What resulted afterwards was an absolute B&M craze, with Inverted coasters being the hot new item for every park to have.  To this day, they remain B&M's most popular coaster design, with at least one being constructed almost every year.

Vekoma - Suspended Looping Coaster (SLC)

The first company to copy B&M's ingenious design was Vekoma, who utilized their basic skeletal-looking track to create compact, cheap, and intense rides.  They utilize inside running wheels and two-seat cars, often combining these cars into massive trains (upwards of 10 cars per train).  However, despite their unique appearance, SLC's are often avoided due to their usual roughness.  The SLC, however, does offer a cheaper alternative to B&M's original Inverted design, a fact which makes it popular with smaller parks looking for a major attraction.  While the basic SLC model exists in great numbers (over 25 clones operate worldwide), the SLC has also been crafted into custom designs, such as Jubilee Odyssey and the Shenlin/Kumali series.
 

 Intamin Suspended Looping Coaster

Intamin also tried their hand at a version of B&M's popular Inverted roller coaster.  Even though their trio of inverted coasters (a lineup which included the world's only complete-circuit launched inverted coaster, Volcano: The Blast Coaster at Paramount's Kings Dominion) were much smoother than Vekoma's, and were indeed raved of as excellent coasters, B&M's perpetual domination of the Inverted coaster market means that Intamin's unique design is often overlooked.  However, Intamin has been moderately successful with a launched shuttle variation of the Inverted coaster - the Impulse coaster, which has been installed at numerous parks worldwide.

Giovanola Inverted

Giovanola produced only one Inverted coaster, the Anaconda in Gold Reef City, Africa.  The coaster is made using the same box-section track that B&M use on their coasters, but features two-across seating (as opposed to B&M's 4-across).  Despite being a very intense and successful ride, Giovanola's Inverted coasters (although still for sale) are, as with Intamin's version, often overlooked due to B&M's domination of the Inverted coaster market.  The fact that their only installation was made in a relatively obscure park don't help the coaster's image.
 

Pinfari Suspended Looping Coaster

Pinfari managed to create a tiny inverted coaster for travelling use.  Marketed under the model name "XP56" and consisting of a tight, compact track layout, two inversions, and 2-across 5-car trains, the ride offers a fairly intense circuit, but suffers from a bit of roughness.  There are two examples in use, one such being the Vortex at M&Ds Scotland's Theme Park.