Novel nonvolatile universal memory technology is essential for providing required storage for nano-computing. As a potential contender for the next-generation memory, the recently found "the missing fourth circuit element", memristor, has drawn a great deal of research interests. In this paper, by starting from basic memristor device equations that assumes constant ion mobility, we develop a comprehensive set of properties and design equations for memristor based memories. Our analyses are specifically targeting key electrical memristor device characteristics relevant to, but not limited to, memory operations. However, like many nano devices, a small voltage drop across the memristor will yield an enormous electric field, which may produce significant highly nonlinear ionic transport that the linear drift assumption no longer holds for realistic memristors. Issues such as how to design circuits facing such nonlinear drift will be discussed. In addition, issues such as how to sense the memory states and perturbations during sensing will be addressed. In this paper, we demonstrate that we can successfully use the derived properties based on the linear drift model to design read and write circuits and analyze important data integrity and noise-tolerance issues for realistic nonlinear drift models.