The despeckling of synthetic aperture radar images using two different convolutional neural network architectures is presented in this paper. The first method presents a novel Siamese convolutional neural network with a dilated convolutional network in each branch. Recently, attention mechanisms have been introduced to convolutional networks to better model and recognize features. Therefore, we propose a novel design for a convolutional neural network using an attention mechanism for an encoder–decoder-type network. The framework consists of a multiscale spatial attention network to improve the modeling of semantic information at different spatial levels and an additional attention mechanism to optimize feature propagation. Both proposed methods are different in design but they provide comparable despeckling results in subjective and objective measurements in terms of correlated speckle noise. The experimental results are evaluated on both synthetically generated speckled images and real SAR images. The methods proposed in this paper are able to despeckle SAR images and preserve SAR features.