Collection and dissemination of data are common tasks motivated by numerous benefits attained through the analysis of rich datasets. Yet many datasets contain sensitive information about individuals which must be duly protected if the data is to be used or shared. Differential privacy is a commonly used disclosure control method for protecting sensitive information while allowing for queries to be posed on databases. The driving idea behind differential privacy is to use a randomization mechanism to add controlled noise to query responses in order to provide a guarantee on the distinguishability between potential configurations of the underlying sensitive data. In this thesis, we focus on two major topics relating to the design and application of differentially private mechanisms. In the first, we focus on the design of mechanisms which employ a range of noisy responses matched to the range of the query posed on the database. Adherence to the range of the query offers the potential for improved utility in a mechanism, yet attaining improved utility in a manner which preserves the differential privacy guarantee is not straightforward. We propose two different approaches to the design of range-adherent mechanisms, one of which is based on the use of a truncated and normalized Laplace distribution, while the other employs linear programming. The second major topic of the thesis covers the application of differential privacy for the obfuscation of facial identity in images. Often, depiction of identity in images may be seen as a breach of privacy, yet the preservation of other information in the images may be desirable. We propose a framework for the application of a distance-based generalization of differential privacy via generative models for images. We provide details on the configuration of a mechanism to achieve a differentially private guarantee in this setting and show how to achieve photo-realistic obfuscated images through the use of various generative models. Within both topics of our work, we implement our proposed approaches and analyze the results of experimental comparisons between our work and other relevant approaches from the literature. We demonstrate improvements in the utility of obfuscated data achieved via our methods.