<p>Nonlinear magnonics harnesses the intricate interactions among magnons (the quanta of spin waves) to create novel states and functionalities for next-generation information processing, surpassing the limitations of linear dynamics in energy efficiency and coherence. This review summarizes recent advances in emerging nonlinear effects induced by three- and four-magnon processes, showing their profound impact on magnon frequencies and wavefunctions across real and momentum spaces. From a semi-classical perspective, we highlight key mechanisms including magnon frequency multiplication via resonant and nonresonant second-harmonic generation, the generation of magnonic frequency combs through magnon-skyrmion and magnon-vortex scattering and phonon-assisted cascades, and the nonlinear topological magnon spin Hall effect induced by emergent gauge fields in spin textures. Transitioning to the quantum regime, we examine how magnon-magnon couplings renormalize energy levels, produce topological band structures by lifting degeneracies, and lead to non-Hermitian phenomena such as exceptional points and lines near Dirac cones. Finally, we spotlight open challenges and opportunities, including nonlinearities in moiré magnets, interaction-driven thermal magnon transport, and magnon stability in altermagnets, envisioning breakthroughs in fundamental spin physics and practical magnonic devices like tunable filters, switches, and quantum sensors.</p>