Abstract: Quasiregular mappings and mappings of finite distortion are natural generalizations of holomorphic mappings to higher dimensions. Whereas the pointwise derivatives of holomorphic mappings map circles to circles, QR-maps and MFD are defined by requiring that the differential maps balls to ellipsoids with controlled eccentricity. Under certain mild integrability conditions, mappings of finite distortion are continuous, open and discrete, as are all quasiregular mappings by the Reshetnyak theorem. For continuous, open and discrete mappings between Euclidean n-domains the branch set, i.e. the set of points where the mapping fails to be a local homeomorphism, has topological dimension of at most $n-2$ by the Cernavskii-Vaisala theorem. For quasiregular mappings more properties for the branch set are known, but several important questions remain open. In this talk we show that an entire mappings of finite distortion cannot have a compact branch set when its distortion is locally finite and satisfies a certain asymptotic growth condition; $K(x) < o(\log (|x|))$. In particular this implies that the branch set of entire quasiregular mappings is either non-compact or empty. We furthermore show that the growth bound is asymptotically strict by constructing a continuous, open and discrete mapping of finite distortion from the Euclidean $n$-space to itself which is piecewise smooth, has a branch set homeomorphic to the$(n-2)$ torus and distortion arbitrarily close to the asymptotic bound $\log (|x|)$. The talk is based on joint work with Aapo Kauranen and Ville Tengvall.