This book is based on a series of lectures given by the author at SISSA,
Trieste, within the PhD courses Techniques in enumerative geometry
(2019) and Localisation in enumerative geometry (2021). The goal of
this book is to provide a gentle introduction, aimed mainly at graduate
students, to the fast-growing subject of enumerative geometry and, more
specifically, counting invariants in algebraic geometry. In addition
to the more advanced techniques explained and applied in full detail to
concrete calculations, the book contains the proofs of several
background results, important for the foundations of the theory. In this
respect, this text is conceived for PhD students or research "beginners"
in the field of enumerative geometry or related areas. This book can be
read as an introduction to Hilbert schemes and Quot schemes on 3-folds
but also as an introduction to localisation formulae in enumerative
geometry. It is meant to be accessible without a strong background in
algebraic geometry; however, three appendices (one on deformation
theory, one on intersection theory, one on virtual fundamental classes)
are meant to help the reader dive deeper into the main material of the
book and to make the text itself as self-contained as possible.