The authors determine or estimate the Bloch constant for normalized \((F (0)=0\) and \(dF(0)=I)\) convex and starlike maps \(F\) of the unit ball \(B_n\) in \(\mathbb{C}^n\) with \(k \)-fold symmetry; that is, \(F(\omega z)= \omega F(z)\) for all \(z\in B_n\), where \(\omega= \exp (2 \pi i/k)\). Among the interesting results established by the authors is that the Bloch constant for normalized univalent convex maps of \(B_n\) with \(k\)-fold symmetry \((k\geq 2)\) is \[ r_k= \int^1_0 {dt \over (1+ t^k)^{2/k}}. \] When \(n\geq 2\) the value of the Bloch constant for the full class of normalized univalent convex maps of \(B_n\) remains unknown but the plausible guess is \(\pi/4\), the exact value when \(n=1\). The Bloch constant for normalized convex maps of the polydisc is \(\pi/4\). Also, for normalized starlike maps of \(B_n\) with \(k\)-fold symmetry the Bloch constant is \(4^{-1/k}\) when \(k=3\) and \(n\geq 4\) and when \(k\geq 4\) and \(n\geq 2\). This is the value of the Koebe constant for such maps.