General features of Beltrami vector fields in a sphere are discussed. Numerical solutions of the kinematic dynamo problem of magnetic fields generation by a Beltrami flow of conducting fluid in a sphere are presented. The Beltrami flow chosen for simulations is chaotic. Maximal temporal growth rates of magnetic field are obtained for magnetic Reynolds numbersRmup to 200, together with associated magnetic eigenmodes forRm100. Their behavior suggests that the dynamo under consideration is fast.