Capability to sequence DNA has been around for four decades now, providing ample time to explore its myriad applications and the concomitant development of bioinformatics methods to support them. Nevertheless, disruptive technological changes in sequencing often upend prevailing protocols and characteristics of what can be sequenced, necessitating a new direction of development for bioinformatics algorithms and software. We are now at the cusp of the next revolution in sequencing due to the development of long and ultra-long read sequencing technologies by Pacific Biosciences (PacBio) and Oxford Nanopore Technologies (ONT). Long reads are attractive because they narrow the scale gap between sizes of genomes and sizes of sequenced reads, with the promise of avoiding assembly errors and repeat resolution challenges that plague short read assemblers. However, long reads themselves sport error rates in the vicinity of 10-15%, compared to the high accuracy of short reads (< 1%). There is an urgent need to develop bioinformatics methods to fully realize the potential of long-read sequencers. Mapping and alignment of reads to a reference is typically the first step in genomics applications. Though long read technologies are still evolving, research efforts in bioinformatics have already produced many alignment-based and alignment-free read mapping algorithms. Yet, much work lays ahead in designing provably efficient algorithms, formally characterizing the quality of results, and developing methods that scale to larger input datasets and growing reference databases. While the current model to represent the reference as a collection of linear genomes is still favored due to its simplicity, mapping to graph-based representations, where the graph encodes genetic variations in a human population also becomes imperative. This dissertation work is focused on provably good and scalable algorithms for mapping long reads to both linear and graph references. We make the following contributions: 1. We develop fast and ...