In conventional multiple-input multiple-output (MIMO), static array configurations struggle in dynamic environments, and further antenna scaling is bounded by cost, energy, and footprint. Emerging approaches, which can enable next-generation wireless communication networks with modest spectrum availability by leveraging flexibility and adaptability rather than sheer array growth, are therefore needed. In this paper, we present a taxonomy framework, referred to as flexible MIMO technology, that systematically categorizes a wide range of evolving MIMO technologies. The focus is on MIMO technologies with flexible physical configurations and integrated applications. We categorize twelve representative flexible MIMO technologies into three major classifications: flexible deployment characteristics-based, flexible geometry characteristics-based, and flexible real-time modifications-based. We then comprehensively overview their fundamental characteristics, potential, and challenges. In addition, we highlight three vital enablers for flexible MIMO technology, including efficient channel state information acquisition schemes, low-complexity beamforming design, and explainable artificial intelligence (AI)-enabled optimization, and discuss eight representative sub-techniques. Finally, two brief case studies -- pre-optimized irregular array for high-speed railway network and cell-free movable antenna -- are presented, showing how flexible MIMO can open new design possibilities and inspire future research directions for next-generation wireless networks.