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We study the theory of Vlasov-Maxwell equilibria in one spatial dimension, aswell as its application to current sheet and flux tube models. The 'inverseproblem' is that of determining a Vlasov-Maxwell equilibrium distributionfunction self-consistent with a given magnetic field. We develop the theory ofinversion using expansions in Hermite polynomials of the canonical momenta.Sufficient conditions for the convergence of a Hermite expansion are found,given a pressure tensor. For large classes of DFs, we prove that non-negativityof the distribution function is contingent on the magnetisation of the plasma,and make conjectures for all classes.  The inverse problem is considered for nonlinear 'force-free Harris sheets'.By applying the Hermite method, we construct new models that can describesub-unity values of the plasma beta $(\beta_{pl})$ for the first time. Whilstanalytical convergence is proven for all $\beta_{pl}$, numerical convergence isattained for $\beta_{pl}= 0.85$, and then $\beta_{pl} = 0.05$ after a're-gauging' process.  We consider the properties that a pressure tensor must satisfy to beconsistent with 'asymmetric Harris sheets', and construct new examples. It ispossible to analytically solve the inverse problem in some cases, others mustbe tackled numerically. We present new exact Vlasov-Maxwell equilibria forasymmetric current sheets, which can be written as a sum of shifted Maxwelliandistributions, ideal for implementations in particle-in-cell simulations.  We study the correspondence between the microscopic and macroscopicdescriptions of equilibrium in cylindrical geometry, and then attempt to findVlasov-Maxwell equilibria for the nonlinear force-free 'Gold-Hoyle' model.However, it is necessary to include a background field, which can bearbitrarily weak if desired. The equilibrium can be electrically non-neutral,depending on the bulk flows.

Theory of one-dimensional Vlasov-Maxwell equilibria: with applications to collisionless current sheets and flux tubes