Monte Carlo simulations and variational calculations using a Gaussian ansatz are applied to a model consisting of a flexible linear polyelectrolyte chain as well as to an intrinsically stiff chain with up to 1000 charged monomers. Addition of salt is treated implicitly through a screened Coulomb potential for the electrostatic interactions. For the flexible model the electrostatic persistence length shows roughly three regimes in its dependence on the Debye-Huckel screening length, kappa(-1). As long as the salt content is low and kappa(-1) is longer than the end-to-end distance, the electrostatic persistence length varies only slowly with kappa(-1). Decreasing the screening length, a controversial region is entered. We find that the electrostatic persistence length scales as root xi(p)/kappa, in agreement with experiment on flexible polyelectrolytes, where xi(p) is a strength parameter measuring the electrostatic interactions within the polyelectrolyte. For screening lengths much shorter than the bond length, the kappa(-1) dependence becomes quadratic in the variational calculation. The simulations suffer from numerical problems in this regime, but seem to give a relationship half-way between linear and quadratic. A low temperature expansion only reproduces the first regime and a high temperature expansion, which treats the electrostatic interactions as a perturbation to a Gaussian chain,, gives a quadratic dependence on the Debye length. For a sufficiently stiff chain, the persistence length varies quadratically with kappa(-1) in agreement with earlier theories.