Toric and Semitoric Symplectic Geometry: Progress and Challenges

Toric integrable systems, also known as symplectic toric manifolds, arise as examples in different contexts within geometry and related areas. Semitoric integrable systems are a generalization of toric integrable systems in dimension four. In this talk I will discuss some classical and recent work on the symplectic geometry of toric and semitoric integrable systems. I will also mention some future challenges in the field.