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In this paper, making use of the truncated Laurent series expansion method and symbolic computation we get the auto-Bäcklund transformation of the (2 + 1)-dimensional Nizhnik–Novikov–Veselov equation. As a result, single soliton solution, single soliton-like solution, multi-soliton solution, multi-soliton-like solution, the rational solution and other exact solutions of the (2 + 1)-dimensional Nizhnik–Novikov–Veselov equation are found. These solutions may be useful to explain some physical phenomena. [ABSTRACT FROM AUTHOR]