The graph width-measure twin-width recently attracted great attention because of its solving power and generality. Many prominent NP-hard problems are tractable on graphs of bounded twin-width if a certificate for the twin-width bound is provided as an input. Bounded twin-width subsumes other prominent structural restrictions such as bounded treewidth and bounded rank-width. Computing such a certificate is NP-hard itself, already for twin-width 4, and the only known implemented algorithm for twin-width computation is based on a SAT encoding. In this paper, we propose two new algorithmic approaches for computing twin-width that significantly improve the state of the art. Firstly, we develop a SAT encoding that is far more compact than the known encoding and consequently scales to larger graphs. Secondly, we propose a new Branch & Bound algorithm for twin-width that, on many graphs, is significantly faster than the SAT encoding. It utilizes a sophisticated caching system for partial solutions. Both algorithmic approaches are based on new conceptual insights into twin-width computation, including the reordering of contractions.