Antimicrobials against bacterial, viral and parasitic pathogens have transformed human and animal health. Nevertheless, their widespread use (and misuse) has led to the emergence of antimicrobial resistance (AMR) which poses a potentially catastrophic threat to public health and animal husbandry. There are several routes, both intrinsic and acquired, by which AMR can develop. One major route is through non-synonymous single nucleotide polymorphisms (nsSNPs) in coding regions. Large scale genomic studies using high-throughput sequencing data have provided powerful new ways to rapidly detect and respond to such genetic mutations linked to AMR. However, these studies are limited in their mechanistic insight. Computational tools can rapidly and inexpensively evaluate the effect of mutations on protein function and evolution. Subsequent insights can then inform experimental studies, and direct existing or new computational methods. Here we review a range of sequence and structure-based computational tools, focussing on tools successfully used to investigate mutational effect on drug targets in clinically important pathogens, particularly Mycobacterium tuberculosis. Combining genomic results with the biophysical effects of mutations can help reveal the molecular basis and consequences of resistance development. Furthermore, we summarise how the application of such a mechanistic understanding of drug resistance can be applied to limit the impact of AMR.