The study of linear oscillations--including simple harmonic, damped, and driven oscillations--is not only fundamental in classical mechanics but lies at the heart of numerous applications in the engineering sciences. Results from research conducted in the context of junior-level mechanics courses suggest the presence of specific conceptual and reasoning difficulties, many of which seem to be based on fundamental concepts. Evidence from pretests (ungraded quizzes) will be presented to illustrate critical difficulties in understanding conceptual underpinnings, relating concepts to graphical representations (e.g., motion graphs), and connecting the physics to the relevant differential equations of motion. Preliminary results from the development of a tutorial approach to instruction, modeled after Tutorials in Introductory Physics by McDermott, et al., suggest that such an approach can be effective in both physics and engineering courses. (Supported by NSF grants DUE-0441426 and DUE-0442388.).