Abstract While reaching and exploring the post-CMOS and beyond-von Neumann era, one thing is for sure; that memristor-based logic design and in-memory computing will attract more and more attention, giving rise to an ever-growing number of different computing approaches. Such approaches may depend either purely on the properties of single devices, or on the collective behaviour of many interconnected devices. In this lecture we present some of the popular trends in memristive logic design, commenting on their most important characteristics, the performance goals, and their overall pros and cons. Moreover, the main focus of recent research efforts on making logic operations as parallel as possible will be also presented. In this context, more in-memory based architectures as well as crossbar-based in-memory parallel processing systems will be discussed. We will focus on novel crossbar-based memristor-transistor ReRAM architectures that enable the integration of logic in-memory by using logic scheme for parallel and single step logic computations. Later, we will enrich the applicability of memristive based computing, coming from a different mostly unconventional angle and inspired by nature. The ability of Physarum polycephalum (slime mould) to learn and adapt to periodic changes in its environment will inspire us to develop behavioral computing memristor-based circuit models. Memristive LC contour and enhanced electronic circuit models of the internal oscillatory motion mechanism of the plasmodium will be studied, aiming to emulate the local signal propagation and the expansion of slime mould vascular network during the reproduction stage, able to solve various complex maze environments. SPICE-level circuit simulations will help the audience to appreciate the efficacy and the advantages of all the presented memristive computing circuits. Bio Dr. Georgios Ch. Sirakoulis (https://gsirak.ee.duth.gr) is Full Professor in the Department of Electrical and Computer Engineering, Democritus University of Thrace, GREECE and he is serving as Visiting Research Professor in University of West England from 2014. Georgios has published more than 110 papers in international journals, all included in the Science Citation Index, more than 140 papers in international conference proceedings and 14 guest-editorials. He is co-editor of 6 books (five international all published by Springer and one national) as well as co-author of 20 book chapters. His research focuses on Complex and Smart Electronic Systems, Future and Emergent Nano-Electronic devices, circuits, models and architectures, Design and Integration of Novel and Emergent Micro-Nano-Bio devices and systems, beyond CMOS computing devices and circuits, Memristors, Bioelectronics and Bioengineering, Unconventional computing, High performance Computing, Cellular Automata Theory & Applications, FPGAs, Modelling and Simulation.