In the organelles of plants and mammals, recent evidence suggests that genomic instability stems in large part from template switching events taking place during DNA replication. Although more than one mechanism may be responsible for this, some similarities exist between the different proposed models. These can be separated into two main categories, depending on whether they involve a single‐strand‐switching or a reciprocal‐strand‐switching event. Single‐strand‐switching events lead to intermediates containing Y junctions, whereas reciprocal‐strand‐switching creates Holliday junctions. Common features in all the described models include replication stress, fork stalling and the presence of inverted repeats, but no single element appears to be required in all cases. We review the field, and examine the ideas that several mechanisms may take place in any given genome, and that the presence of palindromes or inverted repeats in certain regions may favor specific rearrangements.