In the post-Moore era, as the energy consumption of micro-nano electronic devices rapidly increases, near-field radiative heat transfer (NFRHT) with super-Planckian phenomena has gradually shown great potential for applications in efficient and ultrafast thermal modulation and energy conversion. Recently, hyperbolic materials, an important class of anisotropic materials with hyperbolic isofrequency contours, have been intensively investigated. As an exotic optical platform, hyperbolic materials bring tremendous new opportunities for NFRHT from theoretical advances to experimental designs. To date, there have been considerable achievements in NFRHT for hyperbolic materials, which range from the establishment of different unprecedented heat transport phenomena to various potential applications. This review concisely introduces the basic physics of NFRHT for hyperbolic materials, lays out the theoretical methods to address NFRHT for hyperbolic materials, and highlights unique behaviors as realized in different hyperbolic materials and the resulting applications. Finally, key challenges and opportunities of the NFRHT for hyperbolic materials in terms of fundamental physics, experimental validations, and potential applications are outlined and discussed.