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.
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