The physics of astrophysical jets can be divided into three regimes: (i) engine and launch (ii) propagation and collimation, (iii) dissipation and particle acceleration. Since astrophysical jets comprise a huge range of scales and phenomena, practicality dictates that most studies of jets intentionally or inadvertently focus on one of these regimes, and even therein, one body of work may be simply boundary condition for another. We first discuss long standing persistent mysteries that pertain the physics of each of these regimes, independent of the method used to study them. This discussion makes contact with frontiers of plasma astrophysics more generally. While observations theory, and simulations, and have long been the main tools of the trade, what about laboratory experiments? Jet related experiments have offered controlled studies of specific principles, physical processes, and benchmarks for numerical and theoretical calculations. We discuss what has been done to date on these fronts. Although experiments have indeed helped us to understand certain processes, proof of principle concepts, and benchmarked codes, they have yet to solved an astrophysical jet mystery on their own. A challenge is that experimental tools used for jet-related experiments so far, are typically not machines originally designed for that purpose, or designed with specific astrophysical mysteries in mind. This presents an opportunity for a different way of thinking about the development of future platforms: start with the astrophysical mystery and build an experiment to address it.