Complex Langevin (CL) theory of polymer is beyond mean-field simulation

which enables to obtain ensemble averages of physical quantities of polymer systems including fluctuation and correlation effects.As the Hamiltonian of polymer field theory is usually complex

the corresponding Boltzmann factor is non-positive defined.Therefore

instead of the regularly important sampling method

like Monte Carlo

we use the complex Langevin equation to make sampling in the whole complex plane

where a positively defined Boltzmann factor is assumed to be existed.It is shown in this review that complex Langevin simulation is much more efficient than regular sampling methods for studying of polymer systems.Such a method beyond mean-field simulation is very important to study dilute polymer solution

polyelectrolytes systems

in particular

to predict new phases of block copolymer systems

of which mean-field simulations

like self-consistent field theory (SCFT)

have been demonstrated to be not enough in studying these problems.In this review

we give a brief introduction of the complex Langevin simulations.We focus on its theoretical background and the numerical way of how to conduct CL simulations.We show two examples to indicate the advantages of the method beyond mean-field simulation

where fluctuation and correlation effects have been clearly obtained by the CL simulations but missing in SCFT calculations.Complex Langevin simulation has advantages not only in polymer systems but also in ionic systems especially for multi-valence ionic systems.Moreover

developing a method beyond mean-field simulation is also important for biological systems

like understanding the DNA condensation phenomena

the charge inversion inside the virus

and so on.