This book contains expository chapters on Hilbert functions and resolutions. The chapters point out highlights, conjectures, unsolved problems and helpful examples. There has been a lot of interest and intensive research in this area in recent years with important new applications in many different fields, including algebraic geometry, combinatorics, commutative algebra and computational algebra. In ten chapters written by distinguished experts in the area, the book examines the invariant of Castelnuovo-Mumford regularity, blowup algebras and bigraded rings. Then it outlines the current status of the lex-plus-power conjecture and the multiplicity conjecture. It also reviews results on geometry of Hilbert functions and minimal free resolutions of integral subschemes and of equi-dimensional Cohen-Macaulay subschemes of small degree. It also discusses subspace arrangements and closes with an introduction to multigraded Hilbert functions, mixed multiplicities and joint resolutions.