Following recent advances in flexible electronics and programmable metasurfaces, flexible intelligent metasurfaces (FIMs) have emerged as a promising enabling technology for next-generation wireless networks. A FIM is a morphable electromagnetic surface capable of dynamically adjusting its physical geometry to influence the radiation and propagation of electromagnetic waves. Unlike conventional rigid arrays, FIMs introduce an additional spatial degree of design freedom enabled by mechanical flexibility, which can enhance beamforming, spatial focusing, and adaptation to dynamic wireless environments. This added capability enables wireless systems to shape the propagation environment not only through electromagnetic tuning but also through controllable geometric reconfiguration. This article explores the potential of FIMs for next-generation wireless networks. We first introduce the main hardware architectures of FIMs and explain how they can be integrated into wireless communication systems. We then present representative application scenarios, highlighting the advantages of FIMs for future wireless networks and comparing them with other emerging flexible wireless technologies. To illustrate their potential impact, we present case studies comparing FIM-enabled architectures with conventional rigid-array systems, demonstrating the performance gains enabled by surface flexibility for both communication and sensing applications. Finally, we discuss key opportunities, practical challenges, and open research directions that must be addressed to fully realize the potential of FIM technology in future wireless communication systems.