Background The robust model cyanobacterium $Synechocystis$ PCC 6803 is increasingly explored for its potential to use solar energy, water and atmospheric CO$_2$ for the carbon-neutral production of terpenes, the high-value chemicals that can be used for the production of drugs, flavors, fragrances and biofuels. However, as terpenes are chemically diverse, it is extremely difficult to predict whether Synechocystis is a suitable chassis for the photosynthetic production of various terpenes or only a few of them. Results We have performed the first-time engineering and comparative analysis of the best-studied cyanobacterium Synechocystis PCC 6803 for the photosynthetic production of five chemically diverse high-value terpenes: two monoterpenes (C$_{10}$ H$_{16}$ ) limonene (cyclic molecule) and pinene (bicyclic), and three sesquiterpenes (C$_{15}$ H$_{24}$ ) bisabolene (cyclic), farnesene (linear) and santalene (cyclic). All terpene producers appeared to grow well and to be genetically stable, as shown by the absence of changes in their production levels during the 5–9-month periods of their sub-cultivation under photoautotrophic conditions). We also found that $Synechocystis$ PCC 6803 can efficiently and stably produce farnesene and santalene, which had never been produced before by this model organism or any other cyanobacteria, respectively. Similar production levels were observed for cells growing on nitrate (the standard nitrogen source for cyanobacteria) or urea (cheaper than nitrate). Furthermore, higher levels of farnesene were produced by cloning the heterologous farnesene synthase gene in a RSF1010-derived replicating plasmid as compared to the well-used slr0168 neutral cloning site of the chromosome. Conclusions Altogether, the present results indicate that Synechocystis PCC 6803 is better suited to produce sesquiterpenes (particularly farnesene, the most highly produced terpene of this study) than monoterpenes (especially pinene).