This work presents efficient solutions to several basic algorithmic problems regarding periodicity of partial words. In the first part of the paper, we show that all periods of a partial word of length n are determined in O (n log n) time. Moreover, we define algorithms and data structures that help us answer in constant time queries regarding the periodicity of a word’s factors. For these we need an O (n 2) preprocessing time and an O (n) updating time, whenever the words are extended by adding a letter. In the second part of the paper, we show that identifying a way to construct a periodic partial word by substituting the letters on some positions of a full word w with holes, where the distance between two consecutive holes must be greater than a given number, can be done in optimal time O (| w|). We also show that identifying a substitution which replaces the minimum number of positions by holes can be done as fast as in the previous case.