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Monomial ideals form an important link between commutative algebra and combinatorics. Our aim is to study various operations on monomial ideals and algebras, as well as their properties. This thesis consists of an introduction and four research articles. The introduction covers the necessary background and the existing results. The first two papers are devoted to the topic of powers of monomial ideals, their structure and the number of generators. In paper I we investigated the growth of the minimal generating sets. In paper II we studied the so-called Ratliff-Rush closure of monomial ideals of a special kind. Paper III is dedicated to the study of the Lefschetz properties of some interesting classes of monomial and polynomial algebras. In particular, the concept of table ideals was introduced there in order to generalise and simplify some of the previous results on the subject. In paper IV we continued the study of table ideals and used machine learning to distinguish them from non-table ideals.