||High-order finite element methods, using either the continuous or discontinuous Galerkin formulation, are becoming more popular in fields such as fluid mechanics, solid mechanics and computational electromagnetics. While the use of these methods is becoming increasingly common, there has not been a corresponding increase in the availability and use of visualization methods and software that are capable of displaying visualizations of these volumes both accurately and interactively. A fundamental problem with the majority of existing visualization techniques is that they do not understand nor respect the structure of a high-order field, leading to visualization error. Visualizations of high-order fields are generally created by first approximating the field with low-order primitives and then generating the visualization using traditional methods based on linear interpolation. The approximation step introduces error into the visualization pipeline, which requires the user to balance the competing goals of image quality, interactivity and resource consumption. In practice, visualizations performed this way are often either undersampled, leading to visualization error, or oversampled, leading to unnecessary computational effort and resource consumption. Without an understanding of the sources of error, the simulation scientist is unable to determine if artifacts in the image are due to visualization error, insufficient mesh resolution, or a failure in the underlying simulation. This uncertainty makes it difficult for the scientists to make judgments based on the visualization, as judgments made on the assumption that artifacts are a result of visualization error when they are actually a more fundamental problem can lead to poor decision-making. This dissertation presents new visualization algorithms that use the high-order data in its native state, using the knowledge of the structure and mathematical properties of these fields to create accurate images interactively, while avoiding the error introduced by representing the fields with low-order approximations. First, a new algorithm for cut-surfaces is presented, specifically the accurate depiction of colormaps and contour lines on arbitrarily complex cut-surfaces. Second, a mathematical analysis of the evaluation of the volume rendering integral through a high-order field is presented, as well as an algorithm that uses this analysis to create accurate volume renderings. Finally, a new software system, the Element Visualizer (ElVis), is presented, which combines the ideas and algorithms created in this dissertation in a single software package that can be used by simulation scientists to create accurate visualizations. This system was developed and tested with the assistance of the ProjectX simulation team. The utility of our algorithms and visualization system are then demonstrated with examples from several high-order fluid flow simulations.