Invariant theory is a study of the invariant subring of a given ring equipped with a linear group action. Describing the invariant subring was one of the central mathematical problems in the 19th century and many great algebraists such as Cayley, Clebsch, Hilbert, and Weyl had contributed to it. There are many interesting connections between invariant theory and modern birational geometry of moduli spaces. In this talk I will explain some concrete examples including the moduli space of parabolic vector bundles on the projective line and the moduli space of stable rational pointed curves. This talk is based on joint work with Swinarski and Yoo.