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Abstract: In this talk I will discuss symmetries of complex algebraic varieties. When studying a projective variety $X$, one usually wants to understand its symmetries. Conversely, the structure of the group of automorphisms of $X$ encodes relevant geometric properties of $X$. After describing some examples of automorphism groups of projective varieties, I will discuss why the notion of automorphism is too rigid in the scope of birational geometry. We are then led to consider another class of symmetries of $X$, its birational self-maps. The group of birational self-maps of the projective space $\mathbb{P}^n$ is known as the Cremona group in dimension $n$. Describing the structure of the Cremona group is a major problem in algebraic geometry. While the theory is well developed in dimension $2$, little is known in dimension $\geq 3$, and a natural problem is to construct special subgroups of the Cremona group. I will present a recent work with Alessio Corti and Alex Massarenti, where we investigate subgroups of the Cremona group consisting of symmetries preserving some special meromorphic volume forms.