Optimization is ubiquitous in the control of quantum dynamics in atomic, molecular, and optical systems. The ease or difficulty of finding control solutions, which is practically crucial for developing quantum technologies, is highly dependent on the geometry of the underlying optimization landscapes. In this review, we give an introduction to the basic concepts in the theory of quantum optimal control landscapes, and their trap-free critical topology under two fundamental assumptions. Furthermore, the effects of various factors on the search effort are discussed, including control constraints, singularities, saddles, noises, and non-topological features of the landscapes. Additionally, we review recent experimental advances in the control of molecular and spin systems. These results provide an overall understanding of the optimization complexity of quantum control dynamics, which may help to develop more efficient optimization algorithms for quantum control systems, and as a promising extension, the training processes in quantum machine learning.