Automata theory is one of the “dry and boring” upper division classes at a 4-year university. However, there are practical applications. For example, computer language design and implementation depend heavily on the theories related to push down automata (stack machines). Pattern matching in Perl, Awk, Sed and other tools are based on deterministic finite state automata (DFAs).
In electrical and computer engineering, a “state machine” is essentially a DFA, but implemented using logic gates. State machines is an important concept in electrical and computer engineering because they make it easy to express how a device respond to external stimuli.