Magnetic field and impedance of short cylindrical conductors are calculated by Grinberg's method. The analytical solution is derived for the infinitely small and finite electrodes located at the ends of the cylinder. The impedance of short conductors is presented in the form of convergent series. The analytical formulas are validated by the numerical solution of the same problems in alternative formulation. The analytical solution is applied for the calculation of the impedance of a resistive cylinder between two superconductors.