The transmission-disequilibrium test (TDT) of Spielman et al. is a family-based linkage-disequilibrium test that offers a powerful way to test for linkage between alleles and phenotypes that is either causal (i.e., the marker locus is the disease/trait allele) or due to linkage disequilibrium. The TDT is equivalent to a randomized experiment and, therefore, is resistant to confounding. When the marker is extremely close to the disease locus or is the disease locus itself, tests such as the TDT can be far more powerful than conventional linkage tests. To date, the TDT and most other family-based association tests have been applied only to dichotomous traits. This paper develops five TDT-type tests for use with quantitative traits. These tests accommodate either unselected sampling or sampling based on selection of phenotypically extreme offspring. Power calculations are provided and show that, when a candidate gene is available (1) these TDT-type tests are at least an order of magnitude more efficient than two common sib-pair tests of linkage; (2) extreme sampling results in substantial increases in power; and (3) if the most extreme 20% of the phenotypic distribution is selectively sampled, across a wide variety of plausible genetic models, quantitative-trait loci explaining as little as 5% of the phenotypic variation can be detected at the .0001 alpha level with <300 observations.