Height functions are becoming increasingly important in number theory and arithmetic geometry. They provide important arithmetic information about various algebraic objects. I will start with defining height functions on points of a projective space over a number field, polynomial spaces, and Grassmanians, stating some of their main properties, like the theorem of Northcott.
The role of heights in the study of Diophantine approximations and Diophantine equations is well demonstrated by questions about search bounds for solutions of Diophantine equations over a number field, as recently suggested by D.W. Masser. I will review some of the main known results in the field, such as Siegel's lemma, Cassels' theorem on small zeros of quadratic forms, and certain generalizations. Finally, I will talk about some of my recent work on the subject, which can be characterized as an effective approach to the algebraic theory of quadratic forms.