[en] We employ first-principles density-functional calculations to study structural and topological electronic transitions in two-dimensional bismuth layers. Our calculations reveal that a free-standing hexagonal bismuthene phase (the most stable one in the absence of strain) should become thermodinamically unstable against transformation to a putative ‘pentaoctite’ phase (composed entirely of pentagonal and octagonal rings), under biaxial tensile strain. Moreover, our results indicate that 2D bismuth layers in the pentaoctite phase should undergo a topological electronic phase transition under either a biaxial or uniaxial tensile strain. More specifically, at its equilibrium lattice parameters the pentaoctite lattice is a topologically trivial system with a direct band gap. Strain-induced parity inversion of valence and conduction bands is obtained, and the pentaoctite structure undergoes a transition to a topological-insulator phase at a biaxial tensile strain of 5%. In the case of uniaxial tensile strains, the topological transition happens at a tensile strain of 6% along the armchair direction of the pentaoctite lattice, and at a 5% tensile strain in the zigzag direction. Our study indicates that 2D bismuth layers may prove themselves a rich platform to realize topologically non-trivial 2D materials upon strain engineering. (paper)