In the course of this work, we will firstly introduce the concept of computational thinking and provide an overview of the available literature on the topic. Subsequently, we will illustrate some of the most significant initiatives that are being enacted in the United States and in Europe in favor of a didactics of computing science addressed to school-aged children, and will analyze the most popular educational tools used to introduce the students of primary and secondary schools to the basics of computing. As a conclusion, we will provide an overview of the current debate over the role of computational thinking inside primary and secondary education, by analyzing some of the most recent didactic proposals, and suggesting some possible directions for future inquiries.