Bioweapon Beast

The unholy union of Attack Animal and mad science, this is what happens when some genius decides that unpredictable lab-designed killbeasts are far more efficient than those outdated old "tanks" and "fighter jets".

Unless they are in the hands of a Sentai badguy as Monster of the Week, it never ends well. For that exact reason, most of the time this creature type falls under "Awesome but Impractical", but more subdued/plausible versions do exist. Expect Body Horror and Lovecraftian Superpowers out the wazoo. Sister trope to Living Weapon, which is more about versions of conventional weapons made of meat. Often Kaiju if giant sized.

Examples of Bioweapon Beast include:

Anime and Manga

The Guyver suits sort of blur the line between this and Living Weapon, but the Zoanoids and all their variants employed by Chronos definitely fall into this category.
In Neon Genesis Evangelion, this is what the EVAs actually are instead of robots.

Comic Books

Predator X from the X Men series, a creature designed to hunt and kill mutants.

Film

In the Alien series, Prometheus reveals that the Xenomorphs are somehow linked to a mutagenic slime designed by mankind's creators to kill off mankind. Not to mention the ludicrous amount of times Weyland-Yutani has tried to harness them for this purpose.
In the original Piranha, the titular fish. And in the sequel, they crossed them with flying fish.
While there aren't really any in the movies proper, the Star Wars Expanded Universe has a ton of 'em, most of 'em made by the Sith or by the Yuuzhan Vong
Stich technically counts as one of these, given he was created for the sole purpose of destruction. Which, come to think of it, makes this likely the only case of a Girl And Her X plot where the "X" in question is one of these.
The titular Leviathan from the movie of the same name is a Soviet one of these. It is also one part Fish People, one part overly-literal interpretation of a Thomas Hobbes metaphor, and several parts Body Horror.
As revealed in a DVD Special Feature, this is the true origin of the fanged wormlike aliens from The Deadly Spawn.
Sharktopus from the SyFy film of the same name and Pteracuda from the sequel both qualify as these.
The titular creature from the Zeiram films certainly counts, and is considerably smarter than the usual Bioweapon Beast, making it doubly dangerous.

‎Literature

The second Monster Hunter International book mentions that the Manhattan Project briefly and disastrously experimented with these before wisely deciding to go with nuclear weapons instead.

Toys

Cobra's hulking, inhuman Monstro Vipers and Toxo Vipers from G.I. Joe definitely count as this.

Video Games

Resident Evil could easily be called the Trope Codifier for this concept, given how many of the games following this were directly inspired by the horrible things coming out of Umbrella Corp's doors.
Everything spawned by the Blacklight Virus ends up as one of these.
Quite a few of these end up appearing in Final Fantasy VII thanks to the efforts of Professor Hojo.
The zombies from the House of the Dead series are actually much more like this than zombies, especially when you get to the bosses.
The aptly named Biolizard from Sonic Adventure 2 is one of these as it is the prototype of the Ultimate Life Form turned mindless bioweapon against humanity, as well as Gerald Robotnik's Chaos replicas.
Deus Ex features several made by Majestic 12. These include greasels, green reptialian hopping chicken like things that spit acid and karkians, dog like lizards/snapping turtles. It's unclear if grays (standard "gray" aliens) are entirely an example: while initially they seem to be like greasels and karkians some dialog suggests they are merely clones of aliens who died in the Area 51 crash.