Although all meiotic (or linkage) mapping strategies ultimately rely on Mendel's laws, the precise manner in which these laws are exploited in relevant statistical models determines the utility of each strategy for different traits and diseases. In this paper we review the motivation and principles behind each of the three most often used statistical strategies for mapping loci that influence complex, multifactorial traits and diseases (such as diabetes and hypertension), namely: pedigree-based parametric linkage, relative pair allele sharing analysis, and association or linkage disequilibrium analysis. It is our hope to show how Mendel's laws are exploited in each through use of two basic concepts: the short-term evolution of chromosomes, and kinship. Problems inherent in each strategy are described. We then consider how extensions, modifications, or novel derivatives of these three strategies might be fashioned that make better use of the concepts of kinship and short-term chromosome evolution. Two strategies receive emphasis: a haplotype sharing method, which considers the kinship of groups of individuals, and extended variance component models, which make use of genotype information in novel ways.