DNA copies inherited from the same ancestral copy by related individuals are said to be identical by descent (IBD). IBD gives rise to genetic similarities between related individuals. In quantitative genetics, two fundamental problems are heritability estimation and gene mapping for genetic traits. IBD plays a critical role in the study of both problems. When working with population-based samples where pedigree information is unavailable, it is essential to estimate IBD accurately from genetic marker data using pedigree-free methods. The estimated IBD can then be used in heritability estimation and gene mapping using random effects models. For pedigree-free IBD estimation, it is important to use the fact that DNA is inherited in segments as opposed to independent loci. As the single nucleotide polymorphism (SNP) marker panels become increasingly dense, the impact of allelic association (or linkage disequilibrium, LD) on accuracy of IBD estimation also grows. Adjusting for LD in the marker panel can lead to improved IBD estimation accuracy. For heritability estimation and gene mapping using random effects models, a difficult task is to specify the correlation structures of the random genetic effects, which are typically functions of IBD sharing over the putative causal genomic region. Mis-specication of the genetic correlation structures can occur due to inaccurate IBD estimation. This can lead to substantial downward bias in heritability estimation, or loss of power in gene mapping.