Novel or improved microporous battery separator membranes, separators, cells, batteries including such membranes, separators, or cells, and/or methods of making such membranes and/or separators, and/or methods of using such membranes and/or separators. In accordance with at least certain embodiments, an improved or novel battery separator for a secondary or rechargeable lithium battery may have low Electrical resistance of less than 0.95 ohm-cm, or in some cases, less than 0.8 ohm-cm. Furthermore, the inventive battery separator membrane may provide a means to achieve an improved level of battery performance in a rechargeable or secondary lithium battery based on a possibly synergistic combination of low Electrical resistance, low Gurley, low tortuosity, and/or a unique trapezoid shaped pore. In accordance with at least certain multilayer embodiments (by way of example only, a trilayer membrane made of two polypropylene layers with a polyethylene layer in between), the inventive microporous membrane or battery separator may have excellent onset and rate of thermal shutdown performance.