It is an usual practice that any quantum field theory starts with a suitable Lagrangian density. It has been proved enormously successful. I understand, it automatically ensures valuable symmetries of physics to be preserved. But nevertheless the question about the generality of this approach keeps coming to my mind. My question is how one can be sure that this approach has to be always right and fruitful. Isn't it possible, at least from the mathematical point of view that a future theory of physics does not subscribe to this approach?
That's an excellent question, which has a few aspects:


Can you quantize any given Lagrangian? The answer is no. There are classical Lagrangians which do not correspond to a valid field theory, for example those with anomalies.
Do you have field theories with no Lagrangians? Yes, there are some field theories which have no Lagrangian description. You can calculate using other methods, like solving consistency conditions relating different observables.
Does the quantum theory fix the Lagrangian? No, there are examples of quantum theories which could result from quantization of two (or more) different Lagrangians, for example involving different degrees of freedom.


The way to think about it is that a Lagrangian is not a property of a given quantum theory, it also involves a specific classical limit of that theory. When the theory does not have a classical limit (it is inherently strongly coupled) it doesn't need to have a Lagrangian. When the theory has more than one classical limit, it can have more than one Lagrangian description.

The prevalence of Lagrangians in studying quantum field theory comes because they are easier to manipulate than other methods, and because usually you approach a quantum theory by "quantizing" - meaning you start with a classical limit and include quantum corrections systematically. It is good to keep in mind though that this approach has its limitations.
