Keywords: Field theoretic polymer models, flow-structure interaction, mesoscale models, phase field models. Abstract: We will present examples of field theoretic models of multi-component and complex fluids and discuss their main computational challenges and recent advances. We will start with simple phase field based models and progress to a class of field-theoretic models that incorporate exact thermodynamics. In particular, we will present a model for an inhomogeneous melt of elastic dumbbell polymers which incorporates thermodynamic forces acting on the polymers into the hydrodynamic equations. The resulting equations are composed of a system of fourth order PDEs coupled with a nonlinear optimization problem to determine the conjugate fields. We develop a semi-implicit numerical method for the resulting system of PDE's in addition to a parallel nonlinear optimization solver for the conjugate mean-fields. The semi-implicit method effectively removes the fourth order stability constraint associated with explicit methods and we observe a first order time-step restriction. The algorithm for solving the nonlinear optimization problem, which takes advantage of the form of the operators being optimized, reduces the overall computational cost of simulations by several orders of magnitude.