Originating in statistical physics as a model of a porous medium, Bernoulli percolation has become a fundamental model in probability theory. In classical Bernoulli percolation, edges (or vertices) of $\mathbb Z^d$ are deleted independently of each other and with fixed survival probability $p\in[0,1]$. Despite significant progress in understanding this model, basic questions remain, constituting some of the most perplexing problems in probability theory. In the late 1990s, Benjamini, Lyons, Peres, and Schramm initiated a program to study Bernoulli percolation and other invariant percolation models (i.e., random subgraphs whose distribution is invariant under some natural group action) on graphs beyond $\mathbb Z^d$. Cayley graphs of infinite groups provide a rich class of examples, and the behavior of percolations turns out to be closely related to the geometric properties of the underlying group This talk will start with a brief introduction to Bernoulli percolation, highlighting what is known as well as open questions. I will then give a gentle introduction to the aforementioned program, focussing on its main questions and the different motivations behind. I will provide a glimpse into some of the fascinating mathematics involved, primarily by reviewing the case of amenable groups. Finally, I will present recent progress beyond amenability based on joint work with Chiranjib Mukherjee. In the late 1990s, Benjamini, Lyons, Peres, and Schramm initiated a program to study Bernoulli percolation and other invariant percolation models (i.e., random subgraphs whose distribution is invariant under some natural group action) on graphs beyond $\mathbb Z^d$. Cayley graphs of infinite groups provide a rich class of examples, and the behavior of percolations turns out to be closely related to the geometric properties of the underlying group This talk will start with a brief introduction to Bernoulli percolation, highlighting what is known as well as open questions. I will then give a gentle introduction to the aforementioned program, focussing on its main questions and the different motivations behind. I will provide a glimpse into some of the fascinating mathematics involved, primarily by reviewing the case of amenable groups. Finally, I will present recent progress beyond amenability based on joint work with Chiranjib Mukherjee.