A toric variety is a normal variety containing an algebraic torus as an open dense subset whose action on itself extends to the whole space. They provide a fruitful interplay between algebraic geometry and convex geometry since properties on one side (e.g. smoothness, compactness) can be translated into properties involving discrete objects (e.g. cones, fans, polytopes). I will talk about singularities of these varieties from a Hodge theoretic point of view, with applications towards local cohomology and singular cohomology, based on joint works with Sridhar Venkatesh.