The processes that control our climate and the Earth's magnetic field belong to the most complex physical systems that we deal with in Earth Science. Both systems are characterised by fundamental unsolved scientific problems that cannot be explained based on current knowledge, either in terms of observations or in terms of theories. During the recent years emphasis has been on acquiring observations of the needed resolution and accuracy, exemplified by the ESA Swarm constellation mission to measure the geomagnetic field with unprecedented resolution and accuracy. In parallel, much effort has been put into investigations trying to identify whether there are physical processes missing in the current models of climate and geomagnetic field, respectively. In particular scientists have speculated whether there is a physical connection between climate variations and the magnetic field. Some of the unsolved problems in each category will be reviewed and some proposed mechanisms of a physical connection will be discussed.