Area-preserving curvature flows in a two-dimensional homogeneous medium have been studied for several decades. In 1986, Gage showed that an initially convex closed curve remains convex and converges to a circle as time goes to infinity. In many applications in biology and physics, however, the medium is not homogeneous and the objects such as cells and droplets move toward a more favorable environment. As the first step to treat these phenomena, we will consider area-conserving curvature flows in a heterogeneous medium. The properties of the medium are described by a signal function, a smooth function defined in two-dimensional space. The dynamics of curves will be discussed when the areas are small. If time permits, I will also explain the properties of stationary solutions.