We consider all continuous maps of the interval preserving the Lebesgue measure $\lambda$ equipped with the uniform topology. Except for the identity map or $1 - id$ all such maps have topological entropy at least $\log2/2$ and generically they have infinite topological entropy. In this talk we discuss two generic properties: (i) invertibility $\lambda$-a.e. implied by the zero measure-theoretic entropy with respect to $\lambda$, and (ii) complicated structure of level sets. We also recall that there are Besicovitch maps (having no finite or infinite unilateral derivative at any point) preserving $\lambda$ and show that each such map has positive measure-theoretic entropy with respect to $\lambda$.