Klebsiella pneumoniae is one of the most threatening multi-drug resistant pathogens today, with phage therapy being a promising alternative for personalized treatments. However, the intrinsic capsule diversity in Klebsiella spp. poses a substantial barrier to phage host range, complicating the development of broad-spectrum phage-based treatments. Here, we have isolated and genomically characterized phages capable of infecting each of the acquired 77 reference serotypes of Klebsiella spp ., including capsular types widespread among high-risk K. pneumoniae clones causing nosocomial infections. We demonstrated the possibility of isolating phages for all capsular types in the collection, revealing high capsular specificity among taxonomically related phages, in contrast to a few phages that exhibited broad-spectrum infection capabilities. To decipher the determinants of the specificity of these phages, we focused on their receptor-binding proteins, with particular attention to depolymerase domains. We also explored the possibility of designing a broad-spectrum phage cocktail based on phages isolated in reference capsular type strains, and determining the ability to lysate relevant clinical isolates. Interestingly, a combination of 12 phages capable of infecting 60% of the reference Klebsiella spp. serotypes was tested on a panel of carbapenem-resistant K. pneumoniae clinical isolates. Our results suggest that in a highly variable encapsulated bacterial host, phage hunting must be directed to the specific Klebsiella isolates. This work is a step forward in the understanding of the complexity of phage-host interactions, and highlights the importance of implementing precise and phage-specific strategies to treat K. pneumoniae infections worldwide.