A series of studies are presented on the genetic evaluation of cultivated potato (Solanum tuberosum L.) to improve the accuracy and efficiency of selection at various stages of a breeding programme. The central theme was the use of correlated data, such as relationship information and spatial and across-trial correlations, within a linear mixed modelling framework to enhance the evaluation of candidate genotypes and to improve the genetic response to selection. Analyses focused on several social and economically-important traits for the enhancement of the nutritional value, disease resistance and yield of potato tubers. At the formative stages of a breeding scheme, devising a breeding strategy requires an improved understanding of the genetic control of target traits for selection. To guide a strategy that aims to enhance the micronutrient content of potato tubers (biofortification), univariate and multivariate Bayesian models were developed to estimate genetic parameters for micronutrient tuber content from a breeding population generated from crosses between Andean landrace cultivars. The importance of the additive genetic components and extent of the narrow-sense heritability estimates indicated that genotypic 'individual' recurrent selection based on empirical breeding values rather than family-based selection is likely to be the most effective strategy in this breeding population. The magnitude of genetic correlations also indicated that simultaneous increases in important tuber minerals, iron and zinc, could be achieved. Optimising selection efficiency is an important ambition of plant breeding programmes. Reducing the level of candidate replication in field trials may, under certain circumstances, contribute to this aim. Empirical field data and computer simulations inferred that improved rates of genetic gain with p-rep (partially replicated) testing could be obtained compared with testing in fully replicated trials at the early selection stages, particularly when testing over two locations. P-rep testing was able to increase the intensity of selection and the distribution of candidate entries across locations to account for G×E effects was possible at an earlier stage than is currently practised. On the basis of these results, it was recommended that the full replication of trials (at the first opportunity, when enough planting material is available) at a single location in the early stages of selection should be replaced with the partial replication of selection candidates that are distributed over two locations. Genetic evaluation aims to identify genotypes with high empirical breeding values (EBVs) for selection as parents. Using mixed models, spatial parameters to target greater control of localised field heterogeneity were estimated and variance models to account for across-trial genetic heterogeneity were tested for the evaluation of soil-borne powdery scab disease and tuber yield traits at the early stages of a selection programme. When spatial effects improved model fit, spatial correlations for rows and columns were mostly small for powdery scab, and often small and negative for marketable and total tuber yield suggesting the presence of interplot competition in some years for tuber yield traits. For the evaluation of powdery scab, genetic variance structures were tested using data from 12 years of long-term potato breeding METs (multi-environment trials). A simple homogeneous correlation model for the genetic effects was preferred over a more complex factor analytic (FA) model. Similarly, for the MET evaluation of tuber yield at the early stages, there was little benefit in using more complex FA models, with simple correlation structures generally the most favourable models fitted. The use of less complex models will be more straightforward for routine implementation of potato genetic evaluations in breeding programmes. Evaluations for (marketable) tuber yield were extended to multi-location MET data to characterise both genotypes and environments, allowing a re-evaluation of New Zealand MET selection strategies aimed at broad adaptation. Using a factor analytic mixed model, results indicated that the programme’s two main trial locations in the North and the South Islands optimised differentiation between genotypes in terms of G×E effects. There was reasonable performance stability of genotypes across test locations and evidence was presented for some, but limited, genetic progress of cultivars and advanced clonal selections for tuber marketable yield in New Zealand over recent years. The models and selection strategies investigated and developed in this thesis will allow an improved and more systematic application of genetic evaluations in potato selection schemes. This will provide the basis for well informed decisions to be made on selection candidates for the genetic improvement of potato in breeding programmes.