In this talk we review results on several types of harmonic weak Maass forms that are related to integral even weight newforms. We start with a brief introduction to the theory of harmonic weak Maass forms. These can be related to classical modular forms via a certain differential operator, the so-called $\chi $-operator. Starting with an integral weight newform, we will review different constructions of integral weight harmonic weak Maass forms via (generalized) Weierstrass zeta functions that map to the newform under the $\chi $-operator. A second construction via theta liftings gives a half-integral weight harmonic weak Maass form whose coefficients are given by periods of certain meromorphic modular forms with algebraic coefficients and periods of the integer even weight newform. This is joint work with Jens Funke, Michael Mertens, and Eugenia Rosu resp. Jan Bruinier and Markus Schwagenscheidt.[-]