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Solitons are wave packets that can propagate without changing shape by balancing nonlinear effects with the effects of dispersion. In photonics, they have underpinned numerous applications, ranging from telecommunications and spectroscopy to ultrashort pulse generation. Although traditionally the dominant dispersion type has been quadratic dispersion, experimental and theoretical research in recent years has shown that high-order, even dispersion enriches the phenomenon and may lead to novel applications. In this Tutorial, which is aimed both at soliton novices and at experienced researchers, we review the exciting developments in this burgeoning area, which includes pure-quartic solitons and their generalizations. We include theory, numerics, and experimental results, covering both fundamental aspects and applications. The theory covers the relevant equations and the intuition to make sense of the …