Let

be a word. Properties like
``aba is a prefix of u", ``u
contains at least three occurrences of baa", ``u contains an even number
of b's" are local properties.
For infinite words, properties like
``u contains infinitely many
factors of the form babb" are also considered as local properties.
In this survey lecture, I will present results dealing with local
properties of languages and their connection with logic and semigroup
theory.