The nature and power of fixed-point logic with counting

In 1982, Neil Immerman proposed an extension of fixed-point logic by means of counting quantifiers (which we denote FPC) as a logic that might express all polynomial-time properties of unordered graphs. It was eventually proved (by Cai, Fürer and Immerman) that there are polynomial-time graph properties that are not expressible in FPC. Nonetheless, FPC is a powerful and natural fragment of the complexity class PTime. In this article, I justify this claim by reviewing three recent positive results that demonstrate the expressive power and robustness of this logic.