Much of plasma behaviour is governed by multiple distinct nonlinear processes, operating on a wide range of lengthscales and timescales, that are coupled together in innumerable feedback loops. Capturing and quantifying this nonlinear behaviour is crucial at all levels of description, ranging from individual events to global phenomenology. In recent years, a range of techniques derived from complex systems science has been applied successfully to nonlinear plasma datasets. The present paper reviews several of these techniques in the context of applications spanning fusion, space, solar and astrophysical plasmas. Topics include non-Gaussian probability density functions, notably extreme event distributions in fusion and astrophysics and power law distributions in the solar context; differencing and rescaling of fluctuation data, which has yielded information on the number of dominant plasma turbulent processes, and the spatiotemporal ranges over which they operate, in plasmas ranging from microquasar accretion discs to L-mode and dithering H-mode fusion plasmas in the MAST tokamak; quantitative measures of mutual information content and pattern repetition between causally linked but spatiotemporally separated nonlinear events in solar wind and magnetospheric plasmas; global statistics of full-disc solar irradiance; and ELMing, considered as a sequence of pulsed events, in H-mode fusion plasmas in the JET tokamak. These developments in nonlinear plasma data characterization provide fresh additional insights into the underlying plasma physics. They also provide new opportunities for comparing models with data, and with each other, and open avenues for the development of a more rigorous predictive capability in this field.