Abstract: Three-dimensional (3D) printing and bioprinting have come into view for a plannable and standardizable generation of implantable tissue-engineered constructs that can substitute native tissues and organs. These tissue-engineered structures are intended to integrate with the patient’s body. Vascular tissue engineering (TE) is relevant in TE because it supports the sustained oxygenization and nutrition of all tissue-engineered constructs. bioinks have a specific role, representing the necessary medium for printability and vascular cell growth. This review aims to understand the requirements for the design of vascular bioinks. First, an in-depth analysis of vascular cell interaction with their native environment must be gained. A physiological bioink suitable for a tissue-engineered vascular graft (TEVG) must not only ensure good printability but also induce cells to behave like in a native vascular vessel, including self-regenerative and growth functions. This review describes the general structure of vascular walls with wall-specific cell and extracellular matrix (ECM) components and biomechanical properties and functions. Furthermore, the physiological role of vascular ECM components for their interaction with vascular cells and the mode of interaction is introduced. Diverse currently available or imaginable bioinks are described from physiological matrix proteins to nonphysiologically occurring but natural chemical compounds useful for vascular bioprinting. The physiological performance of these bioinks is evaluated with regard to biomechanical properties postprinting, with a view to current animal studies of 3D printed vascular structures. Finally, the main challenges for further bioink development, suitable bioink components to create a self-assembly bioink concept, and future bioprinting strategies are outlined. These concepts are discussed in terms of their suitability to be part of a TEVG with a high potential for later clinical use.