In this talk, we discuss the question of Gibbs point processes in $\mathbb{R}^d$ with pairwise interactions that are not integrable at infinity. A standard example is the Riesz potential of the form $\varphi(x)=\frac{1}{|x|^s}$ where $s<d$. This setting has a long history, notably because the case $s=d-2$ corresponds to the classical Coulomb potential, which arises from electrostatic theory. We will first address the existence of the process in the infinite volume regime when a neutralizing background is introduced (this model is known as Jellium in theoretical physics). Subsequently, we will discuss the rigidity of such point processes, specifically hyper-uniformity and number rigidity. We will provide a state-of-the-art review and present numerous conjectures and open problems.