In this paper, we study the problem of sorting unichromosomal linear genomes by prefix double-cut-and-joins (or DCJs) in both the signed and the unsigned settings. Prefix DCJs cut the leftmost segment of a genome and any other segment, and recombine the severed endpoints in one of two possible ways: one of these options corresponds to a prefix reversal, which reverses the order of elements between the two cuts (as well as their signs in the signed case). Depending on whether we consider both options or reversals only, our main results are: (1) new structural lower bounds based on the breakpoint graph for sorting by unsigned prefix reversals, unsigned prefix DCJs, or signed prefix DCJs; (2) a polynomial-time algorithm for sorting by signed prefix DCJs, thus answering an open question in [8]; (3) a 3/2-approximation for sorting by unsigned prefix DCJs, which is, to the best of our knowledge, the first sorting by {\em prefix} rearrangements problem that admits an approximation ratio strictly smaller than 2 (with the obvious exception of the polynomial-time solvable problems); and finally, (4) an FPT algorithm for sorting by unsigned prefix DCJs parameterised by the number of breakpoints in the genome.