A transuranium element is any chemical element that comes after Uranium in the periodic table. These elements have atomic numbers greater than 92. All transuranium elements are radioactive, and none are found in nature except Neptunium and Plutonium. An element that is radioactive will decay into other elements.
From what we do know about the transuranium elements, as the atomic number goes up, the more radioactive it gets. Some isotopes have very long half-lives, such as Americium-243 having a half life of 7370 years. Others do not, like Tennessine-294 which has a half life of 51 milliseconds.
The applications of them are used far and wide, like how Uranium and Plutonium are used for fuel and weapons, Curium is used for rovers, Berkelium is used for creating other elements, notably Tennessine, and Americium being used for smoke detectors. Fermium to Lawrencium really don't have uses and after that, what scientists call superheavy elements are Rutherfordium to Oganesson, which also have no uses.