Modern high-tech machines have to be designed to withstand extreme conditions, like near vacuum or high Mach number flows. Common to these scenarios is the breakdown of standard fluid models. The kinetic description by the Boltzmann equation offers a unified approach to gas dynamics. Posed as scalar PDE on the six-dimensional phase space it provides a challenging problems owing to its high-dimensional, nonlinear collision operator. Conserving its continuous translational and rotational invariance is key for any numerical method. Solutions of the Boltzmann equation can possess sharp local features that demand a locally refined basis. However, naive grid-based approaches will likely fail to preserve the continuous symmetries of the PDE on the discrete level. We show how to construct invariant compactly supported basis functions of arbitrary order and how to exploit their special structure for the fast assembly of the nonlinear collision operator. Its invariance properties allow us to drastically reduce the computational complexity of the arising high-dimensional integrals in the weak form. Finally, we discuss preconditioners for the Jacobian of the discrete residual that take advantage of the translational invariance to split the six-dimensional problem into two three-dimensional problems. We close the talk with remarks on the high performance aspects of implementations in modern multistage programming languages.