  In this thesis, we explore the critical phenomena in the presence of extended
objects, which we call defects, aiming for a better understanding of the
properties of non-local objects ubiquitous in our world and a more practical
and realistic study of criticality. To this end, we study the statistical
O$(N)$ vector model in $(4-\epsilon)$ dimensions with three kinds of defects: a
line defect constructed by smearing an O$(N)$ vector field along one direction
and Dirichlet and Neumann boundaries. A conventional approach to critical
phenomena would be to perform perturbative calculations using Feynman diagrams
and doing renormalization group analysis. But we here also take a different but
complementary approach based on three axioms that include conformal symmetry of
the theory at the criticality. We apply this axiomatic framework to the
critical O$(N)$ model with a defect and reproduce the perturbative results at
the leading non-trivial order in $\epsilon$, substantiating the validity of our
approach. Along the way, we develop and refine the axiomatic framework to
derive anomalous dimensions of the composite operators on the defect that have
not been accessible in the existing literature by focusing on the analyticity
of the correlation functions.
