The irreversibility constraint, the non-convexity of governing energy functional and the intrinsically small length-scale are the main sources of algorithmic and numerical challenges for phase-field models of fracture. The talk aims at summarizing the main ideas, results and challenges that we proposed and encountered in addressing the above issues in the past few years. We highlight - various solution strategies for the discretized coupled problem, such as partitioned (staggered) and frontal (monolithic) schemes, with a particular focus on their robustness and efficiency, - various options of incorporating the crack irreversibility constraint, with special focus on our newly proposed penalization approach with a practical and accurate bound for the penalty constant, - a posteriori estimation analysis for the discretization error and the induced adaptive mesh refinements, with a specified hierarchy of the “adapt” and “solve” processes. With intensive benchmarking, the implications of the above on simulation results are illustrated and discussed. This is a joint work with L. De Lorenzis.