integrable systems are, roughly, dynamical systems with many conserved quantities. recently, pelayo-v\~{u} ng\d{o}c classified semitoric integrable systems, which generalize toric integrable systems in dimension four, in terms of five symplectic invariants. using this classification, i construct a metric on the space of semitoric integrable systems. by studying continuous paths in this space produced via symplectic blowups i determine its connected components. this uses a new algebraic technique in which i lift matrix equations from $\mathrm{sl}(2,\mathbb{z})$ to its preimage in the universal cover of $\mathrm{sl}(2,\mathbb{r})$ and i further use this technique to completely classify all semitoric minimal models. i also produce invariants of integrable systems by constructing an equivariant version of the ekeland-hofer symplectic capacities and, as a first step towards a metric on general integrable systems, i provide a framework to study convergence properties of families of maps between manifolds which have distinct domains. this work is partially joint with \'alvaro pelayo, daniel m. kane, and alessio figalli.