  The subject of our thesis is the uniqueness theory of meromorphic functions
and it is devoted to problems concerning Bruck conjecture, set sharing and
related topics. The tool, we used in our discussions is classical Nevanlinna
theory of meromorphic functions. In 1996, in order to find the relation between
an entire function with its derivative, counterpart sharing one value CM, a
famous conjecture was proposed by R. Bruck. Since then the conjecture and its
analogous results have been investigated by many researchers and continuous
efforts have been put on by them. In our thesis, we have obtained similar types
of conclusions as that of Bruck for two differential polynomials which in turn
improve several existing results under different sharing environment. A number
of examples have been exhibited to justify the necessity or sharpness of some
conditions, hypothesis used in the thesis. As a variation of value sharing, F.
Gross first introduced the idea of set sharing, by proposing a problem, which
has later became popular as Gross Problem. Inspired by the Gross' Problem, the
set sharing problems were started which was later shifted towards the
characterization of the polynomial backbone of different unique range sets. In
our study, we introduced some new type of unique range sets and at the same
time, we further explored the anatomy of these unique range sets generating
polynomials as well as connected Bruck conjecture with Gross' Problem.
