The present review shows a summary of current trends in high-fidelity simulations of turbulent flows in moderately complex geometries. These trends are put in the historical context of numerical simulations, starting with early weather predictions and continuing with seminal direct-numerical-simulation work. Here we discuss high-fidelity simulations conducted in a number of complex geometries, including ducts, pipes, wings and obstacles, and describe the potential of the spectral-element method (SEM) to carry out such simulations. Finally, we provide a number of future directions where novel data-driven methods can exploit the great wealth of highquality turbulence data in the literature.