The review summarises recent theoretical achievements and observational manifestations of a new, recently discovered type of nonlinear oscillations in multi-component plasmas, namely super-nonlinear periodic waves and super-nonlinear solitary waves (supersolitons). Both are characterised by a non-trivial topology of their phase portraits, highly anharmonic profile shapes, extremely long periods, and large amplitudes. Based upon multi-fluid magnetohydrodynamic plasma models, examples of ion-acoustic and Alfvén super-nonlinear waves are considered. A multi-component nature of the plasma was revealed to be a crucial condition for the existence of these super-nonlinear waves, with the complexity of the system growing with the number of plasma species accounted for in the model. A minimum number of plasma components which allow for the existence of supernonlinear waves are also discussed. From the observational point of view, typical signatures of periodic super-nonlinear waves are manifested, for example, in the oscillatory processes operating in the magnetised plasma of the solar corona and ground-based plasma machines. Super-nonlinear solitary structures (supersolitons) of an electrostatic origin are recognised in the Earth's magnetosphere, laboratory experiments with chemically active plasmas, and numerical simulations.