Planetary systems are not only composed of planets, but also of km-sized rocky and icy bodies that are confined within belts similar to the Asteroid and Kuiper belt in the Solar System. Mutual collisions within these belts grind down solids producing dust and giving rise to debris discs. Primitive asteroids and comets likely played a major role in the emergence of life on Earth through their delivery of volatiles early in the lifetime of our planet. Cometary impacts, therefore, could be a necessary condition for the emergence of life in exoplanets and the study of debris discs essential to determine the ubiquity of such phenomenon. Moreover, exocometary discs provide a unique window into the origins and outer regions of planetary systems as comets do within our Solar System. Initially, in Chapter 1 I present an overview of the study of exoplanetary systems, focusing on debris discs. I discuss the basics of planet formation, its connection with debris discs, and how these evolve and interact with planets. I also describe how we observe these discs and probe their volatile component that is locked inside exocomets, and some evidence supporting the idea of exocomets venturing into the inner regions of planetary systems. Then, in Chapters 2, 3, 4 and 5 I present new ALMA observations of the systems HD 181327, η Corvi, the multiplanet system 61 Vir and HD 107146, which host debris discs. In the first two, I highlight the derivation of the density structure of their discs and the detection of volatiles being released by exocomets; while in the third and fourth I compare the observations with simulations, which I use to set constraints on the underlying planetesimal distribution and mass and orbital distance of unseen planets. Finally, in Chapter 6 I present result obtained from N-body simulations to study the process of inward transport of comets by a multiplanetary system and how these can deliver material to inner planets and explain the frequently observed exozodiacal dust. To conclude, in Chapter 7 I summarise the results and conclusions of this dissertation and discuss ongoing and future work.