While co-addition and subtraction of astronomical images stand at the heart of observational astronomy, the existing solutions for them lack rigorous argumentation, are not achieving maximal sensitivity and are often slow. Moreover, there is no widespread agreement on how they should be done, and often different methods are used for different scientific applications. I am going to present rigorous solutions to these problems, deriving them from the most basic statistical principles. These solutions are proved optimal, under well defined and practically acceptable assumptions, and in many cases improve substantially the performance of the most basic operations in astronomy. For coaddition, we present a coadd image that under the assumption of spatially uniform noise is: a) sufficient for any further statistical decision or measurement on the entire data set. b) improves both survey speed (by 5-20%) and effective spatial resolution of ground based astronomical surveys c) improves substantially imaging through turbulence applications such as lucky imaging and speckle interferometry d) much faster than many of the currently used coaddition solutions. For subtraction, we present a subtraction image that is: a) Free of subtraction artifacts, hopefully relieving the transient detection pipelines from machine learning algorithms and human scanning. b) optimal for transient detection under the assumption of spatially uniform noise. c) sufficient for any further statistical decision including the identification of cosmic rays and other image artifacts. d) orders of magnitude faster than existing subtraction methods. e) allows accurate statistical analysis of the resulting subtraction image, allowing exact knowledge of a transients significance.