The Internet of Things (IoT) has become an essential part of our society. IoT devices are used in our houses, hospitals, cars, industry, etc., making our lives easier. Nonetheless, there are a number of serious concerns about security, privacy and performance issues in IoT. It has been proven that the aforementioned issues are strictly related to the high degree of centralisation of current IoT architecture. Thus, there is an increasing interest in adopting blockchain in IoT. However, blockchain adoption is not straightforward due to the power, storage and computational limitations of IoT. Consequently, the concept of lightweight blockchain is getting more and more attention from researchers and engineers. In this paper, we conduct a systematic literature review on the lightweight blockchain concept for IoT following the PRISMA methodology. We systematically analyse "lightweight blockchain for IoT" proposals in order to better understand the limitations of blockchain for IoT, the characteristics of the current work on this topic and further research opportunities. Specifically, we analyse the definition of lightweight blockchain that other authors give, the characteristics of the reviewed proposals, their "lightweight" aspects and their evaluation. Finally, we discuss the results of the review along with further research opportunities. Consequently, this work is mostly focused on understanding the technical and performance-related aspects of blockchain for IoT as a prelude to more specific analysis such as security (i.e., attacks, vulnerabilities, etc.).