What are the applications of automata theory?

 DFA as a model of computation
 I’ve done some theoretical research on formal language theory and DFAs, so my immediate response was why DFAs are important to theorists.

DFAs and regular expressions

Regular expressions are useful tool that every programmer should know. If you wanted to check if a string is a valid email address, you might write something like:
/^([a-z0-9_\.-]+)@([\da-z\.-]+)\.([a-z\.]{2,6})$/
DFAs in compilers
In every programming language, the first step in the compiler or interpreter is the lexer. The lexer reads in a file of your favorite programming language, and produces a sequence of tokens.

DFAs for artificial intelligence
Another application of finite automata is programming simple agents to respond to inputs and produce actions in some way. You can write a full program, but  a DFA is often enough to do the job. DFAs are also easier to reason about and easier to implement.

DFAs in probability
What if we took a DFA, but instead of fixed transition rules, the transitions were probabilistic? This is called a Markov Chain!