Massive multiple-input multiple-output (MIMO) has been a critical enabling technology in 5th generation (5G) wireless networks. With the advent of 6G, a natural evolution is to employ even more antennas, potentially an order of magnitude more, to meet the ever-increasing demand for spectral efficiency. This is beyond a mere quantitative scale-up. The enlarged array aperture brings a paradigm shift towards near-field communications, departing from traditional far-field approaches. However, designing advanced transceiver algorithms for near-field systems is extremely challenging because of the enormous system scale, the complicated channel characteristics, and the uncertainties in the propagation environments. Hence, it is important to develop scalable, low-complexity, and robust algorithms that can efficiently characterize and leverage the properties of the near-field channel. In this article, we discuss the principles and advocate two general frameworks to design deep learning-based near-field transceivers covering both iterative and non-iterative algorithms. Case studies on channel estimation and beam focusing are presented to provide a hands-on tutorial. Finally, we discuss open issues and shed light on future directions.