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A Bloch function g is a function analytic in the unit disk such that (1 ; j zj 2 ) jg 0 (z)j is bounded. First we generalize the theorem of Rohde that, for every \bad" Bloch function, g(r ) (r ;! 1) follows any prescribed curve at a boundeddistance for in a set of Hausdor dimension almost one. Then we introduce balanced Bloch functions. They are characterized by the fact that jg 0 (z)j does not vary much on each circle fjzj = rg except for small exceptional arcs. We show e.g. that Z 1 0 jg 0 (rdoi:10.4171/rmi/261 fatcat:pbrfhv3jjffhhj44ecxvmwj364