Abstract.Image registration, in particular medical image registration, has been subject to extensive 

studies in the past years. Its versatile and important applications have attracted researchers 

from various branches, including numerical analysts. In this note, we not only give an 

overview on some new and exiting approaches but also point out some of the mathemati- 

cal challenges associated with these approaches. Moreover, we explain why variational based 

methods to image registration naturally lead to large scaled and challenging optimization and 

hence numerical linear algebra problems. We present various instructive examples and com- 

ment on numerical schemes.