Look at the list of
logic-related conferences. If you include smaller events that are related to logic, but not necessarily have mathematical logic as the main focus, they happen around the world almost every week.
Most of them are related to computer science. Logic turned out to be a wonderful tool for building models and investigating the properties of software and hardware. It's one of the main foundations of
formal methods. So even though in the beginning of the 20th century logic was one of the most abstract areas of mathematics, now it is one of the most applied ones.
One of the most direct applications of proof theory is formal verification of hardware and software as well as of regular mathematical results. A great deal of research is devoted to making proving properties of programs convenient and, hopefully, not significantly harder than writing programs themselves.
Logic can be applied to other areas of mathematics. For example, using compactness theorem, it is easy to show that every partial order can be extended to a linear one, and that every field $F$ has an extension $F'$ where every polynomial with coefficients from $F$ has a root. Satisfiability solvers can be used to solving computationally complex combinatorial problems, such as finding shortest routes that visit all cities. I am not a specialist about connection with physics, but there are some, such as
quantum logic.