Homozygosity/Autozygosity mapping is a powerful technique, but is only valid when searching for a mutation segregating within a small, closed population.  Such a small population, possibly created by the founder effect, will have a limited gene pool, and thus any inherited disease will probably be a result of two copies of the same mutation segregating on the same haplotype.  Since affected individuals will probably be homozygous in the regions, looking at SNPs in a region is an adequate marker of regions of homozygosity and heterozygosity.  Modern day SNP arrays are used to survey the genome and identify large regions of homozygosity.  Homozygous blocks in the genomes of affected individuals can then be laid on top of each other, and the overlapping region should contain the disease gene.
This analysis is often extended by analyzing autozygosity, an extension of homozygosity, in the genomes of affected individuals.  This can be accomplished by plotting a cumulative LOD score alongside the overlaid blocks of homozygosity.  By taking into consideration the population allele frequencies for all SNPs via autozygosity mapping, the results of homozygosity can be confirmed.  Furthermore, if two suspicious regions appear as a result of homozygosity mapping, autozygosity mapping may be able to distinguish between the two (ex. If one block of homozygosity is a result of a very non-diverse region of the genome, the LOD score will be very low).
Tools for Homozygosity Mapping
# HomSI: a homozygous stretch identifier from next-generation sequencing data  A tool that identifies homozygous regions using deep sequence data.