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Mode-division multiplexing (MDM) technology has drawn tremendous attention for its ability to expand the link capacity within a single-wavelength carrier, paving the way for large-scale on-chip data communications. In the MDM system, the signals are carried by a series of higher-order modes in a multi-mode bus waveguide. Hence, it is essential to develop on-chip mode-handling devices. Silicon-on-insulator (SOI) has been considered as a promising platform to realize MDM since it provides an ultra-high-index contrast and mature fabrication processes. In this paper, we review the recent progresses on silicon integrated nanophotonic devices for MDM applications. We firstly discuss the working principles and device configurations of mode (de)multiplexers. In the second section, we summarize the multi-mode routing devices, including multi-mode bends, multi-mode crossings and multi-mode splitters. The inverse-designed multi-mode devices are then discussed in the third section. We also provide a discussion about the emerging reconfigurable MDM devices in the fourth section. Finally, we offer our outlook of the development prospects for on-chip multi-mode photonics.