Graphs can naturally model complex systems such as chemical molecules or social networks. However, their complex and irregular structure makes learning from graphs a challenging and fascinating area of research. In this talk, I will explore the intersection of graph representation learning and homomorphism counting—a technique to measure how frequently specific patterns occur in graphs. Homomorphism counting has emerged as a powerful tool in the study and development of graph neural networks (GNNs). I will focus on two areas: (1) Understanding and quantifying what GNNs can do and where they fail and (2) improving the capabilities of GNNs. Special interest in both these areas lies on the expressivity of GNNs, i.e., their ability to learn different representations for nonisomorphic graphs. I will present and discuss the basics, as well as recent results in this interesting and active area of research.