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.