This paper focuses on the potential accuracy of remote sensing (RS) image registration. We investigate how this accuracy can be estimated without ground truth available and used to improve registration quality of mono- and multimodal pair of images. At the local scale of image fragments, the Cramer-Rao lower bound (CRLB) on registration error is estimated for each local correspondence between coarsely registered pair of images. This CRLB is defined by local image texture and noise properties. Opposite to the standard approach, where registration accuracy is only evaluated at the output of the registration process, such valuable information is used by us as an additional input knowledge. It greatly helps in detecting and discarding outliers and refining the estimation of geometrical transformation model parameters. Based on these ideas, a new area-based registration method called registration with accuracy estimation (RAE) is proposed. In addition to its ability to automatically register very complex multimodal image pairs with high accuracy, the RAE method is able to provide registration accuracy at the global scale as a covariance matrix of estimation error of geometrical transformation model parameters or as pointwise registration standard deviation. This accuracy does not depend on any ground truth availability and characterizes each pair of registered images individually. Thus, the RAE method can identify image areas for which a predefined registration accuracy is guaranteed. This is essential for RS applications imposing strict constraints on registration accuracy such as change detection, image fusion, and disaster management. The RAE method is proved successful with reaching subpixel accuracy while registering eight complex mono-/multimodal and multitemporal image pairs including optical-to-optical, optical-to-radar, optical-to-digital elevation model (DEM) images, and DEM-to-radar cases. Other methods employed in comparisons fail to provide in a stable manner accurate results on the same test cases.