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Haplotype phasing and imputation have become essential components of genome-wide association analysis pipelines, as these methods allow imputation of genetic variation from smaller whole-genome sequenced reference panels into larger cohorts (genotyped much more sparsely at low cost). Over the past two decades, phasing and imputation methods have undergone several generations of development as sample sizes and variant counts in typical analyses have each increased by five orders of magnitude. I will overview the algorithmic themes that have emerged from these approaches -- many based on the Li-Stephens hidden Markov model -- and discuss the computational considerations that are now informing future directions in this field.