Interactively Deforming a Mesh?

Is there a way to interactively deform a mesh like using the controller to distort Mario’s face in the beginning of Super Mario 64? If not with unreal, any idea how it could be done?

Mario’s face was something to do with bones or morph targets.
There are also ways to deform meshes using a height map, but you have to know a lot of things about materials and shaders to do it.