The rapid growth in the number of interconnected devices on the Internet (referred to as the Internet of Things—IoT), along with the huge volume of data that are exchanged and processed, has created a new landscape in network design and operation. Due to the limited battery size and computational capabilities of IoT nodes, data processing usually takes place on external devices. Since latency minimization is a key concept in modern-era networks, edge servers that are in close proximity to IoT nodes gather and process related data, while in some cases data offloading in the cloud might have to take place. The interconnection of a vast number of heterogeneous IoT devices with the edge servers and the cloud, where the IoT, edge, and cloud converge to form a computing continuum, is also known as the IoT-edge-cloud (IEC) continuum. Several key challenges are associated with this new computing systems’ architectural approach, including (i) the design of connection and programming protocols aimed at properly manipulating a huge number of heterogeneous devices over diverse infrastructures; (ii) the design of efficient task offloading algorithms aimed at optimizing services execution; (iii) the support for security and privacy enhancements during data transfer to deal with the existent and even unforeseen attacks and threats landscape; (iv) scalability, flexibility, and reliability guarantees to face the expected mobility for IoT systems; and (v) the design of optimal resource allocation mechanisms to make the most out of the available resources. These challenges will become even more significant towards the new era of sixth-generation (6G) networks, which will be based on the integration of various cutting-edge heterogeneous technologies. Therefore, the goal of this survey paper is to present all recent developments in the field of IEC continuum systems, with respect to the aforementioned deployment challenges. In the same context, potential limitations and future challenges are highlighted as well. Finally, indicative use cases are also presented from an IEC continuum perspective.