Technological advances in three-dimensional (3D) cell culture have reshaped both basic research and translational applications in biomedicine. Unlike conventional two-dimensional (2D) systems, which cannot reproduce tissue architecture, biochemical gradients, or complex cell–cell and cell–matrix interactions, 3D platforms better recapitulate the native microenvironment and physiological dynamics that drive differentiation, gene expression, and drug responses. Within this family of models, spheroids—relatively simple, homogeneous multicellular aggregates—serve as practical and cost-effective tools for drug screening and cancer biology, whereas organoids derived from differentiated stem cells capture higher-order organotypic structure and function, making them especially valuable for disease modeling and precision medicine. Emerging technologies such as 3D bioprinting and organ-on-a-chip systems extend these capabilities by enabling precise architectural control, programmable microenvironments, and dynamic physiological cues under in vitro conditions. This review provides a critical, integrated analysis of principal 3D culture classes, compares their strengths and limitations, and examines recent progress in 3D bioprinting and organ-on-a-chip technologies with particular emphasis on bioink engineering, functional vascularization, and strategies to promote tissue maturation. By synthesizing current evidence, we outline a practical framework to guide the translation from simple laboratory constructs toward robust, human-relevant tissue models for research and therapeutic development.