We survey recent results on the structure of the range of the derivative of a smooth mapping $f$ between two Banach spaces $X$ and $Y$ . We recall some necessary conditions and some sufficient conditions on a subset $A$ of $\mathcal{L}(X,Y)$ for the existence of a Fréchet differentiable mapping $f$ from $X$ into $Y$ so that $f'(X)=A$ . Whenever $f$ is only assumed Gâteaux differentiable, new phenomena appear: for instance,
there exists a mapping $f$ from $\ell^{1}(\mathbb{N})$ into $\mathbb{R}^{2}$ , which is bounded, Lipschitz-continuous, and so that for all $x,y\in \ell^{1}(\mathbb{N})$ , if $x\ne y$ , then $\Vert f'(x)-f'(y)\Vert >1$ .