In this dissertation recent advances in modeling various atomic and molecular systems with quantum mechanical calculations employing explicitly correlated Gaussian functions are presented. The author has utilized multiple approaches and considered a number of approximations to develop optimal calculation frameworks. Electronic and muonic molecules and atoms have been considered. A number of unique calculations have been performed and some novel and interesting results, including high accuracy description of the charge asymmetry in the heteronuclear systems and lifetimes of rotationless vibrational levels of diatomic molecules, have been generated.