Let $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$. Let $X$ be a normed space over $\mathbb{F}$. It is said that $x\in X$ is Birkhoff-James orthogonal to $y\in X$, $$x\perp y\quad \text{if}\quad \|x+\lambda y\|\geq\|x\|\quad \text{for all }\lambda\in\mathbb{F}.$$ For inner product spaces, this definition is equivalent to the usual inner product orthogonality. We shall say that a mapping $f$ (not necessarily linear) between two normed spaces $X$ and $Y$ over $\mathbb{F}$ preserves Birkhoff-James orthogonality in both directions if $$x \perp y \, \Longleftrightarrow \, f(x) \perp f(y) \quad \text{for all }x,y \in X.$$ Let $(H,(\cdot,\cdot))$ and $(K,(\cdot,\cdot))$ be inner product spaces over $\mathbb{F}$ and suppose that $f \colon H\to K$ is a mapping satisfying the so-called Wigner equation $$\vert (f(x),f(y)) \vert = \vert (x,y) \vert\quad \text{for all } x,y\in H.$$ Obviuosly, if $f$ satisfies the Wigner equation then it preserves inner product orthogonality in both directions, but the converse is not true. The celebrated Wigner's unitary-antiunitary theorem says that $f$ satisfies the Wigner equation if and only if $$f(x)=\sigma(x) Ux\quad \text{for all } x\in H,$$ for some mapping $\sigma$ from $H$ to modulus one scalars and a linear or an antilinear isometry $U \colon H \to K$. Studying of inner product orthogonality preservers begins with the Uhlhorn's proof of Wigner’s theorem. There are several ways of generalizing Wigner's theorem and the Wigner equation to normed spaces. The aim of this talk is to present some generalizations of Wigner's theorem to a certain class of normed spaces, to characterize nonlinear Birkhoff-James orthogonality preservers and to relate them with a generalized Wigner equation.