Due to unsustainable increase in total power consumption of the modern information technology field, the conventional evolution of the computer based on the von Neumann computing architecture should be reconsidered, and it is time to seek a feasible replacement for or supplement to the current computing paradigm. In this review, memristors are examined from the frameworks of both von Neumann and neuromorphic computing architectures. Memristor was suggested to be present as the fourth circuit element that correlates the charge and flux by Chua in 1971. However, it had been largely obsolete until the Hewlett Packard research group claimed that the previously known resistance switching (RS) TiO2 was actually a memristor in 2008. For the von Neumann computer, a new logic computational process based on the material implication is discussed. It consists of several memristors which play roles of combined logic processor and memory, called stateful logic circuit. In this circuit configuration, the logic process flows primarily along a time dimension, whereas in current von Neumann computers it occurs along a spatial dimension. In the stateful logic computation scheme, the energy required for the data transfer between the logic and memory chips can be saved. In addition, the memory in this circuit is basically non-volatile so energy required for the data refresh is also saved. Neuromorphic computing refers to a computing paradigm that mimics the human brain which is also called cognitive computing. Currently, the neuromorphic or cognitive computing mainly relies on the software emulation of several brain functionalities, such as image and voice recognition utilizing the recently highlighted deep learning algorithm. However, the human brain typically consumes ~10 - 20 Watts for selected “human-like” tasks, which can be currently mimicked by a supercomputer with power consumption of several tens of kilo- to mega-Watts. Therefore, hardware implementation of such brain functionality must be eventually sought for power-efficient computation. Several fundamental ideas for utilizing the memristors and their recent progresses in these regards are reviewed. Finally, material and processing issues are dealt with, which is followed by the conclusion and outlook of the field.