Gallery models in buildings provide combinatorial models to approach geometric or representation theoretic problems involving reductive groups and their loop groups. In this talk I will introduce this framework in the context of Kazhdan-Lusztig theory, showing how they help to compute Kazhdan-Lusztig R polynomials and understand their combinatorics. In order to treat similar problems for loop groups of Kac-Moody groups (which are infinite dimensional analogs of reductive groups), I will present a continuous generalization involving masures, a Kac-Moody analog to Bruhat-Tits buildings introduced by Gaussent and Rousseau in 2008.