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We discuss various questions of the following kind: for a continuous map $X\to Y$ from a compact metric space to a simplicial complex, can one guaranteethe existence of a fiber large in the sense of Urysohn width? The $d$-widthmeasures how well a space can be approximated by a $d$-dimensional complex. Theresults of this paper include the following.  1) Any piecewise linear map $f: [0,1]^{m+2} \to Y^m$ from the unit euclidean$(m+2)$-cube to an $m$-polyhedron must have a fiber of $1$-width at least$\frac{1}{2\beta m +m^2 + m + 1}$, where $\beta = \sup_y \text{ rk }H_1(f^{-1}(y))$ measures the topological complexity of the map.  2) There exists a piecewise smooth map $X^{3m+1} \to \mathbb{R}^m$, with $X$a riemannian $(3m+1)$-manifold of large $3m$-width, and with all fibers beingtopological $(2m+1)$-balls of arbitrarily small $(m+1)$-width.

Waist of maps measured via Urysohn width