The Internet of Bodies (IoB) is an imminent extension to the vast Internet of things domain, where interconnected devices (e.g., worn, implanted, embedded, swallowed, etc.) located in-on-and-around the human body form a network. Thus, the IoB can enable a myriad of services and applications for a wide range of sectors, including medicine, safety, security, wellness, entertainment, to name but a few. Especially considering the recent health and economic crisis caused by novel coronavirus pandemic, a.k.a. COVID-19, the IoB can revolutionize today’s public health and safety infrastructure. Nonetheless, reaping the full benefit of IoB is still subject to addressing related risks, concerns, and challenges. Hence, this survey first outlines the IoB requirements and related communication and networking standards. Considering the lossy and heterogeneous dielectric properties of the human body, one of the major technical challenges is characterizing the behavior of the communication links in-on-and-around the human body. Therefore, this paper presents a systematic survey of channel modeling issues for various link types of human body communication (HBC) channels below 100 MHz, the narrowband (NB) channels between 400 MHz and 2.5 GHz, and ultra-wideband (UWB) channels from 3 to 10 GHz. After explaining bio-electromagnetics attributes of the human body, physical and numerical body phantoms are presented along with electromagnetic propagation tool models. Then, the first-order and the second-order channel statistics for NB and UWB channels are covered with a special emphasis on body posture, mobility, and antenna effects. For capacitively, galvanically, and magnetically coupled HBC channels, four different channel modeling methods (i.e., analytical, numerical, circuit, and empirical) are investigated, and electrode effects are discussed. Lastly, interested readers are provided with open research challenges and potential future research directions.