In computer graphics realm, three-dimensional graphical data, generally represented by triangular meshes, have become commonplace, and are deployed in a variety of application processes (e.g., smoothing, compression, remeshing, simplification, rendering, etc.). However, these processes inevitably introduce artifacts, altering the visual quality of the rendered 3D data. Thus, in order to perceptually drive the processing algorithms, there is an increasing need for efficient and effective subjective and objective visual quality assessments to evaluate and predict the visual artifacts. In this thesis, we first present a comprehensive survey on different sources of artifacts in digital graphics, and current objective and subjective visual quality assessments of the artifacts. Then, we introduce a newly designed subjective quality study based on evaluations of the local visibility of geometric artifacts, in which observers were asked to mark areas of 3D meshes that contain noticeable distortions. The collected perceived distortion maps are used to illustrate several perceptual functionalities of the human visual system (HVS), and serve as ground-truth to evaluate the performances of well-known geometric attributes and metrics for predicting the local visibility of distortions. Our second study aims to evaluate the visual quality of texture mapped 3D model subjectively and objectively. To achieve these goals, we introduced 116 processed models with both geometric and texture distortions, conducted a paired-comparison subjective experiment, and invited 98 subjects to evaluate the visual qualities of the models under two rendering protocols. Driven by the collected subjective opinions, we propose two objective visual quality metrics for textured meshes, relying on the optimal combinations of geometry and texture quality measures. These proposed perceptual metrics outperform their counterparts in term of the correlation with the human judgment.