In this paper, a novel continuous-aperture array (CAPA)-based wireless communication architecture is proposed, which relies on an electrically large aperture with a continuous current distribution. First, an existing prototype of CAPA is reviewed, followed by the potential benefits and key motivations for employing CAPAs in wireless communications. Then, three practical hardware implementation approaches for CAPAs are introduced based on electronic, optical, and acoustic materials. Furthermore, several beamforming approaches are proposed to optimize the continuous current distributions of CAPAs, which are fundamentally different from those used for conventional spatially discrete arrays (SPDAs). Numerical results are provided to demonstrate their key features in low complexity and near-optimality. Based on these proposed approaches, the performance gains of CAPAs over SPDAs are revealed in terms of channel capacity as well as diversity-multiplexing gains. Finally, several open research problems in CAPA are highlighted.