Automata are used in various ways..
In computer science:-
1. DFA/NFA - used for Lexical Analysis in Compilers
2. Moore/Mealy - concept is used in Digital CIrcuits, precisely Sequential Logic.
3. CFG and CSG - used in Parsing and Other compiler constructs such as type checking, etc.
4. LBA - The whole real world computer model can be seen as an LBA (i.e. TM with restricted input tape)
5. TM - concepts are used in theoretical computer science extensively.
Other applications:-
https://cs.stanford.edu/people/eroberts/courses/soco/projects/2004-05/automata-theory/apps.html
