Memristors, as new electronic elements, have been under rigorous study in recent years, owing to their good memory and switching properties, low power consumption, nano-dimensions and a good compatibility to present integrated circuits, related to their promising applications in electronic circuits and chips. The main purpose of this paper is the application and analysis of the operations of metal–oxide memristors in logic gates and complex schemes, using several standard and modified memristor models and a comparison between their behavior in LTSPICE at a hard-switching, paying attention to their fast operation and switching properties. Several basic logic gates—OR, AND, NOR, NAND, XOR, based on memristors and CMOS transistors are considered. The logic schemes based on memristors are applicable in electronic circuits with artificial intelligence. They are analyzed in LTSPICE for pulse signals and a hard-switching functioning of the memristors. The analyses confirm the proper, fast operation and good switching properties of the considered modified memristor models in logical circuits, compared to several standard models. The modified models are compared to several classical models, according to some significant criteria such as operating frequency, simulation time, accuracy, complexity and switching properties. Based on the basic memristor logic gates, a more complex logic scheme is analyzed.