Images captured underwater are often characterized by low contrast, color distortion, and noise. To address these visual degradations, we propose a novel scheme by constructing an adaptive color and contrast enhancement, and denoising (ACCE-D) framework for underwater image enhancement. In the proposed framework, Gaussian filter and Bilateral filter are respectively employed to decompose the high-frequency and low-frequency components. Benefited from this separation, we utilize soft-thresholding operation to suppress the noise in the high-frequency component. Accordingly, the low-frequency component is enhanced by using an adaptive color and contrast enhancement (ACCE) strategy. The proposed ACCE is a new adaptive variational framework implemented in the HSI color space, in which we design a Gaussian weight function and a Heaviside function to adaptively adjust the role of data item and regularized item. Moreover, we derive a numerical solution for ACCE, and adopt a pyramid-based strategy to accelerate the solving procedure. Experimental results demonstrate that our strategy is effective in color correction, visibility improvement, and detail revealing. Comparison with state-of-the-art techniques also validate the superiority of propose method. Furthermore, we have verified the utility of our proposed ACCE-D for enhancing other types of degraded scenes, including foggy scene, sandstorm scene and low-light scene.