Signal denoising is one of the classical problems in the area of signal processing. Noise models can be categorized as additive and multiplicative. This chapter considers problems such as inpainting as denoising problems. There are classical techniques based on filtering for removing all kinds of signal noise. This chapter focuses on sparsity-based techniques for removing noise. It discusses Gaussian denoising which is the most popular technique for removing impulse noise. The basic assumption behind sparsity-based denoising is that the signal can be sparsely represented in a transform domain but the noise cannot. If this assumption holds, one can always find a threshold (in the transform domain) that suppresses the dense but small noise and keeps the sparse high-valued signal components. After noise suppression, the denoised signal is obtained by applying the inverse transform.