We analyze nonlinear properties of microstructured materials with the negative refractive index, the so-called left-handed metamaterials. We demonstrate that the hysteresis-type dependence of the magnetic permeability on the field intensity allows changing the material properties from left to right handed and back. Using the finite difference time domain simulations, we study the wave reflection from a slab of a nonlinear left-handed material and observe generation and propagation of temporal solitons in such materials. We demonstrate also that the nonlinear left-handed metamaterials can support both transverse electric-- and transverse magnetic--polarized self-trapped localized beams, spatial electromagnetic solitons. Such solitons appear as single-hump and multihump beams, being either symmetric or antisymmetric, and they can exist because of the hysteresis-type magnetic nonlinearity and the effective domains of negative magnetic permeability.