Anomaly mining is an important problem that finds numerous applications in various real world domains such as environmental monitoring, cybersecurity, finance, healthcare and medicine, to name a few. In this article, I focus on two areas, (1) point-cloud and (2) graph-based anomaly mining. I aim to present a broad view of each area, and discuss classes of main research problems, recent trends and future directions. I conclude with key take-aways and overarching open problems.