The programming paradigm that focuses on functions and their applications is called functional programming. This approach prioritizes the use of functions as the fundamental building blocks of software design. Rather than focusing on mutable data and state, functional programming emphasizes immuable data and pure functions, which produce outputs based solely on input values without causing side effects. Popular languages that support functional programming include Haskell, Scala, and functional aspects of JavaScript, benefiting developers with easier reasoning about code behavior.