We present recent developments on a posteriori error estimation and adaptive algorithms for Computational Fluid Dynamics (CFD), with particular emphasis on turbulent flow, complex geometry, coupled problems and high performance computing. Turbulent flow poses challenges both with regards to the cost of resolving fine scale turbulent structures, in the bulk of the flow as well as near solid boundaries, and with regards to predictability of quantities of interest as turbulent flows show chaotic features. We review of our work towards a parameter-free method for simulation of turbulent flow at high Reynolds numbers, where we develop a model for turbulent flow in the form of weak solutions of the Navier–Stokes equations, approximated by an adaptive finite element method, where: (i) viscous dissipation is assumed to be dominated by turbulent dissipation proportional to the residual of the equations, and (ii) skin friction at solid walls is assumed to be negligible compared to inertial effects. The result is a computational model without empirical data, where the only model parameter is the local size of the finite element mesh. Under adaptive refinement of the mesh based on a posteriori error estimation, output quantities of interest in the form of functionals of the finite element solution converge to become independent of the mesh resolution, and thus the resulting method has no adjustable parameters. No ad hoc design of the mesh is needed, instead the mesh is optimized based on solution features, in particular no boundary layer mesh is needed. We connect the computational method to the mathematical concept of a dissipative weak solution of the Euler equations, as a model of high Reynolds number turbulent flow, and we highlight a number of benchmark problems for which the method is validated.