Narrow-line Seyfert 1 galaxies (NLS1s) are a subclass of active galactic nuclei (AGN) identified more than 30 years ago, but still not entirely understood. Their most distinctive feature is the narrowness of their permitted lines, which is not interpreted as a sign of obscuration as in other AGN, but is instead attributed to low rotational velocity around a relatively low mass black hole. Since their luminosity is comparable to that of typical broad-line Seyfert 1 galaxies (BLS1s), their Eddington ratio is high, suggesting that these objects are characterized by fast-accreting black holes. Interestingly enough, some of them have been detected in gamma-rays, a sign that they can harbor powerful relativistic jets. In this seminar I will outline their main properties in the framework of the quasar main sequence, and discuss how the presence of relativistic jets in them affects our understanding of jet physics. I will also show how many of their properties indicate that their true nature may be that of young AGN in a recently triggered activity phase, and how they could be connected with high-redshift quasars. Finally, I will report some recent results showing that a new, elusive population of absorbed relativistic jets not detected at low radio frequencies may be hiding among these intriguing objects.