The aim of this thesis was to investigate some aspects of breeding for quantitative traits in apple. First, this study explored the measurement and inheritance of a complex quantitative trait (fruit shape). Fourier analysis was used to mathematically describe apple fruit shape in an objective manner and principal component analysis grouped the Fourier descriptors into meaningful shape traits. Heritabilities were estimated to determine genetically inherited shape traits, and genotype by environment interactions were estimated to determine the stability of the trait expression across environments. Fruit aspect ratio accounted for over 76% of the phenotypic variation in shape, fruit conicity for 6% and fruit squareness for only 2%. These traits had moderate to high narrow-sense heritabilities (0.79, 0.38 and 0.38 respectively), which indicated that individual selection would be efficient. High between-site genetic correlation (rA > 0.8), indicated low genotype-by-environment interaction and suggests breeding based at one site would be efficient for altering shape at other sites. A shape chart was constructed to enable these three apple shape traits to be evaluated quickly and accurately in the field, avoiding the need for image capture and Fourier analysis. Second, heritabilities and genetic correlations were estimated for a range of apple traits from a genetically broad-based apple breeding population of 71 families. Narrow-sense heritability was estimated at each of two sites and across sites for fruit acidity (0.17-0.22), fruit sugars (0.25-0.26), and tree growth habit (0.19-0.41). Higher heritabilities were estimated for fruit squareness (0.32-0.43), fruit conicity (0.32-0.46), powdery mildew incidence (0.40), vigour (0.28-0.62), fruit firmness (0.44-0.53), harvest time (0.66-0.82), leafing (0.60-0.83), fruit aspect ratio (0.74-0.89), flowering (0.92) and fruit size (0.90-1.01). Traits with a high narrow-sense heritability could be selected for efficiently with ind