The authors define and investigate convexity structures in the sense of Takahashi in T-0-quasi-metric spaces. They prove that numerous important results about convexity structures in metric spaces can be generalized to the quasi-metric setting. They also show that the latter convexity structures naturally occur in asymmetrically normed real vector spaces and in q-hyperconvex T-0-quasi-metric spaces. In the T-0-quasi-metric setting they explore various interesting additional conditions that convexity structures in the sense of Takahashi can satisfy. (C) 2015 Elsevier B.V. All rights reserved.