Polynomial and moment optimization problems are infinite dimensional optimization problems that can model a wide range of problems such as shape-constrained polynomial regression, optimal control of dynamical systems, region of attraction, polynomial matrix decomposition, smooth maximum-likelihood density estimation, AC power systems, experimental design, and computation of Nash equilibria. In this minisymposium we show how the Julia and JuMP ecosystems are particularly well suited for constructing and solving these problems. In particular, we show how the JuMP extensions SumOfSquares/PolyJuMP allow for an effortless construction of these problems and how they provide a flexible and customizable building block for additional packages such as JuliaMoments. We also show how various features of the Julia programming language are used in the state-of-the-art solvers Hypatia.jl and Aspasia.jl. Finally, we showcase specific uses of these tools for applications in engineering and statistics.