I introduce the history of Feynman diagrams origin and show importance of Feynman representation for the development of approximation free Diagrammatic Monte Carlo (DMC) method. Outlining essential features of the DMC method I highlight our developments and applications of this approach to various systems with electron-phonon and electron-magnon interactions. I show that the DMC method is strongly enhanced by our Stochastic Optimization Consistent Constraints (SOCC) method of analytic continuation which paves the way to approximation free information on the dynamical properties (ARPES, optical conductivity, mobility, dynamical structure factor of neutron scattering, etc.).

Showing the applications of the classical DMC to various systems I emphasize its inability to treat the many-fermion systems because of the famous fermion sign problem. I outline the method of Bold DMC (BDMC), developed to circumvent the sign problem, show some results of its application to many-polaron systems, and outline directions of further development to obtain the first-time approximation free results for the many-body systems with electron-phonon and Coulomb interactions.

I show that Popov-Fedotov trick gives possibility to apply BDMC to spin systems and demonstrate how exact and comprehensive can be treatment of the Heisenberg model. Finally, I show that the BDMC method is applicable to artificial Kondo systems whose fabrications and studies started to boom recently. I show that the BDMC method is capable of comprehensive treatment of these systems including calculation of a plenty of its experimental properties with no approximations: ARPES, optical conductivity, mobility, magnetic susceptibility, etc.