Inverse problems arise in many areas of image processing, where the goal is to recover unknown quantities from indirect, incomplete, or noisy measurements. These problems are typically ill-posed, meaning that small changes in the data can lead to large changes in the solution, or that solutions may not exist or be unique. To overcome these challenges, variational regularization methods have become essential. In this chapter, we introduce the foundational concepts of inverse problems, discussing their mathematical formulation, including examples from image deblurring and denoising. We then present variational regularization as a powerful framework to stabilize these problems by incorporating prior information about the solution, such as smoothness or sparsity. The chapter explores classical regularization techniques like Tikhonov regularization and Total Variation (TV), along with more advanced models, and examines their mathematical properties (like convexity) and practical applications. We also discuss the role of regularization parameters, the construction of energy functionals by their physical properties and the corresponding Euler-Lagrange equations, which serve as the basis for designing efficient numerical algorithms. Through a combination of theory and examples, this chapter provides a comprehensive overview of how variational regularization methods address inverse problems in image processing.