Besides getting killed, the Predators are known to get captured and their technology stolen in many cases throughout the franchise. Here is a list of the most notable cases of captive Predators, with many of them being experimented on and most of them managing to escape in spectacular fashion.
Alpha Predator - Enslaved By The Amengi
The Yautja race had its origins tied to being captives and slaves to another race - the Amengi. The Hish were ancestors of Yautja who were used as laborers and fighters for entertainment. An Alpha Predator rose through the ranks of the Hish and started a rebellion on Yautja Prime, leading to a revolution that overthrew their former masters. He decorated his armor and bio-mask with the style of the Amengi insectoid bodies, having an exoskeleton and large head with yellow eyes. The Hish then assimilated a lot of the Amengi technology into their own, including devices like the shoulder cannon and cloak. The Alpha (also known as the First Hunter) became a legend among the Yautja and many wanted to become worthy hunters in his honor.
Area 52 Predator
The Area 52 Predator was captured along with this ship and held in the secret base of Colonial Marines. Another unnamed Predator was sent in after him, to help him escape and to bring back his captured ship and equipment. Unfortunately, he was too late, as the captured Predator was already experimented on by humans, impregnated with a facehugger a gave birth to a Predalien. Titled the abomination, the Predalien was killed by the unnamed Predator and the Predator's honor was restored. This triggered a self destruct of Area 52 and the whole base exploded after the unnamed Predator escaped in the ship. Both Predators appeared in Rebellion's Aliens vs. Predator 1999 PC game. You don't explicitly see the captured Predator, only his cell, his blood, and the Predalien.
The Castaway Predator arrived in the radioactive wasteland of the Cursed Earth near Mega-City one in the Predator Versus Judge Dredd Versus Aliens: Splice and Dice comic book. Instead of having an enjoyable hunt, he became hunted himself by the genetically enhanced minions of Dr. Niels Reinstöt. Castaway was gravely injured and brought to the labs of the mad scientist. In addition to himself, his ship was also captured along with the trophies of Xenomorphs. With the help of an Alien skull, Dr. Reinstöt was able to clone a Xenomorph egg and acquire a facehugger. The facehugger was attached to the Castaway who birthed a Predalien, killing him. He was later avenged by his 4 clanmates and Judge Dredd, who killed the Predalien and Dr. Reinstöt. Castaway was one of the few Predators who was unable to escape from his captors.
Stone Heart was one of the three Predators captured by the Borgia Industries, along with Long Spear and Swift Knife, in the Predator: Concrete Jungle video game. He came to Neonopolis with his brothers to hunt for worthy prey, but was attacked by Hunter Borgia and the Monster Squad. He was captured and put into stasis in the company labs, experimented on, aga finally made into a cyborg with enhanced strength. Later, Stone Heart was released by his captors to hunt the Scarface Predator who had become a nuisance for them. Scarface gained the upper hand in the fight and Stone Heart was killed, giving him an honorable death and releasing his spirit.
The Fugitive Predator was captured and held by Project Stargazer during his escape to Earth from the Upgrade Predator. Some of his gear was stolen by Quinn McKenna even before he was captured. At the Stargazer base, he was strapped to a table and restrained from his arms and legs. However, he had no problems escaping when the humans were distracted by the appearance of the Upgrade Predator ship. He wrecked havoc in the base, even using some human weapons to kill Stargazer soldiers. The Fugitive Predator escaped the base successfully but was killed later by the Upgrade Predator. It is is interesting to note that another Predator was held at Stargazer base and escaped in similar circumstances in the Predator: Hunters and Hunted prequel book. The Fugitive Predator actually acquired his mask when escaping. Seemed that Stargazer was really bad at security.
The Thrill of the Hunt Predator
The Thrill of the Hunt Predator was a survivor of a Predator hunt gone wrong. His clanmates were all killed by Xenomorphs and the lone Predator detonated his self-destruct device while managing to survive the blast by falling into a chasm. While injured and unconscious, he was captured by the colonist of the planet. He endured torturous tests and examinations while planning his escape. When the humans thought they had pumped enough intoxicating gas in the room and went to check on him, he grabbed a scientist as a human shield and escaped. A human hunting party was sent after him, but most of them died. He reclaimed his equipment at an ancient template and was finally killed by one of the main characters of the comic book.
The Crucified Predator from the Predators movie was a classic Predator tied up by the Super Predator clan on the Game Preserve Planet to a totem pole. He was one of the few Predators known to be captured and held by other Predators, being a victim of a clan war between the different Predator types. His capture is depicted in the motion comic titled Crucified which was released with the Predators Blu-ray and DVD. Later he was freed by Royce and fought the Berserker Predator. Being weakened by his long imprisonment, he lost the fight and was beheaded. His ship was destroyed by the Berserker Predator as well.
Dachande was one of the most famous Predators from the initial Aliens vs. Predator comic book series. Upon reaching Ryushi with his clan, his hunt met with disaster almost immediately. A human landspeeder crashed into their Predator lander ship, making it explode and rendering Dachande unconscious. Subsequently, he was captured by humans and brought to their colony, titled Prosperity Wells. Strapped to a table, he was examined by the humans who theorized on his physiology and cell structure. Later, in the confusion of the Xenomorph outbreak and the attack of the bad-blood Tichinde, Dachande escaped from his restraints. He killed Tichinde, fought a horde of Xenomorphs together with Machiko Noguchi and finally succumbed to his injuries.
Big Mama was a notorious female Predator and one of the few of her kind. While being drawn to Caryn Delacroix and hunting her, she was incapacitated and captured by the security forces of Lucien Delacroix, the husband of Caryn. An EMP device was used to short out her equipment and she lost consciousness. Like for most other captured Predators, they conducted tests on her and were amazed by her physiology. Caryn was in return drawn implicitly to Big Mama and helped to free her and to return her equipment. Together they escaped and became hunting buddies, fighting Xenomorphs, robots and white hybrids.
Prince was one of the most badass Predators ever, but still he was captured during his hunt on LV-1201. General Vassili Rykov set a trap for him in the forward observation pods and used an EMP to short out Prince's weapons and equipment. He was captured and observed for a short time, with scientists trying to detach his wrist gauntlets unsuccessfully. Prince was accidentally released by a power surge caused by the escaping Xenomorphs. He reacquired his weapons and gear, escaped the base and later had his revenge on Rykov by extracting his spine. Prince appeared as the main playable Predator character in Monolith's Aliens vs. Predator 2 PC game.
The Captive Predator
Probably the most prominent captive Predator was held in a biodome in the Nevada desert by humans in the near future. He lost his right arm along with the wrist-gauntlet during the capture. He was not much hindered by it, as his left arm was abnormally big and strong with giant razor-sharp claws. The humans thought they had the situation under control, but actually, that was far from the truth. The Predator figured out their security system and was always one step ahead of them. In the end, he faked his own death, killed his captors and reacquired his wrist gauntlet. He made the whole base explode by his self-destruct device. His final fate is unknown, but it is unlikely he survived. This all occurred in the acclaimed Predator: Captive comic book.