  Let $S \subset R$ be an arbitrary subset of a unique factorization domain $R$
and $\K$ be the field of fractions of $R$. The ring of integer-valued
polynomials over $S$ is the set $\mathrm{Int}(S,R)= \{ f \in \mathbb{K}[x]:
f(a) \in R\ \forall\ a \in S \}.$ This article is an effort to study the
irreducibility of integer-valued polynomials over arbitrary subsets of a unique
factorization domain. We give a method to construct special kinds of sequences,
which we call $d$-sequences. We then use these sequences to obtain a criteria
for the irreducibility of the polynomials in $\mathrm{Int}(S,R).$ In some
special cases, we explicitly construct these sequences and use these sequences
to check the irreducibility of some polynomials in $\mathrm{Int}(S,R).$ At the
end, we suggest a generalization of our results to an arbitrary subset of a
Dedekind domain.
