Motivation: Although modern high-throughput biomolecular technologies produce various types of data, biosequence data remain at the core of bioinformatic analyses. However, computational techniques for dealing with this data evolved dramatically. Results: In this bird's-eye review, we overview the evolution of main algorithmic techniques for comparing and searching biological sequences. We highlight key algorithmic ideas emerged in response to several interconnected factors: shifts of biological analytical paradigm, advent of new sequencing technologies and a substantial increase in size of the available data. We discuss the expansion of alignment-free techniques coming to replace alignment-based algorithms in large-scale analyses. We further emphasize recently emerged and growing applications of sketching methods which support comparison of massive datasets, such as metagenomics samples. Finally, we focus on the transition to population genomics and outline associated algorithmic challenges.