The subject of this talk will be the theoretical properties and the efficiency of a family of penalization methods for the construction of diffusion processes with weak constraints. Broadly these processes refer to a class of dynamics whose -time-marginal or path- distribution (time-marginal or path) is conditioned to satisfy a given property. Such processes appear in a variety of problems from stochastic mechanics to the modeling of insider trader, crowd modeling (among other examples) and more widely to constrained stochastic control problem. The first part of the talk will be dedicated the particular case of diffusion processes whose time -marginal distributions are restrained to lie in a given set. In this case, a simple and efficient penalization approximation, which extends classical penalization methods for reflected SDEs, can be built. The second part of the talk will be dedicated to discussing penalization methods for constrained stochastic control problems. In this situation, penalization can be found efficient  for a certain class of constraints, but, a particular example - relating to matching problem- reveal some seemingly original feature, that will be conversed.   Part of this presentation is based on a current work with Kerlyns Martinez, Universidad de Valparaiso.