A language is called palindromic if each of its words is a palindrome. The palindromicity problem for a family \({\mathcal L}\) of languages is the problem of deciding whether or not a given language L in \({\mathcal L}\) is palindromic. The authors study this problem and show that it is positively answered in the case of regular, context-free and HDT0L languages. Additionally, the characterization results for families of regular and context-free palindromic languages are presented.