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Several features of sample surveys generally render inapplicable the st and ard explicit forms of variance estimation. The often complex nature of the estimators of the population parameters used, the complexities of sample design, and the large number of estimates usually presented have led practitioners to seek general methods of estimating sampling errors. A number of variance estimation techniques have been developed to address these problems, but the properties of these methods are not fully understood. In this dissertation analytic methods are used to investigate the properties of several versions of the replicated variance estimation procedures of balanced repeated replication and jackknifing. Expressions for the approximate variance and bias of a number of replicated variance estimation procedures are derived in Chapters 2, 3, and 4. These are used to provide guidelines for choosing the replicated procedure most suitable for a given survey. One procedure, in particular, termed the Combined Strata Grouped Jackknife, is recommended. This flexible procedure provides good control over the precision and cost of variance estimation, while generally being subject to only slight bias. In Chapter 5 the variance and bias of the collapsed strata procedure for estimating variances when only one PSU is selected per stratum are considered. Conditions under which the optimal collapsing is in pairs, triples, or more are studied. It is concluded that collapsing in pairs is generally to be preferred except when only a few strata are involved. The choice as to which strata to pair is also considered. Chapter 6 discusses a method of generalizing variance estimates for a whole population to its constituent crossclasses in multistage samples. Linear models are used to provide theoretical justification for the methods proposed. An assumption of a constant rate of homogeneity across crossclasses is found to be often satisfactory for categorical variables, although a minor modification sometimes give improvement. An assumption of a constant rate of homogeneity may be frequently inappropriate for continuous variables, and an adjustment is required if substantial bias in crossclass variance estimation is to be avoided.