Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of invariants and designing methods and algorithms to compute them remains an active area of ongoing research with an abundance of applications, in particular, to mathematical physics, computer image processing and recognition, computer-aided design, and biological modeling. In this incredibly vast topic, I will focus on displaying a fruitful interplay between the methods from differential and algebraic invariant theories.