We report the first AlOx-based resistive switching memory (RRAM) using carbon nanotubes (CNT) as contact electrodes. CNTs with average diameter of 1.2nm effectively localize the conduction filaments (CFs). The Al/AlOx/CNT device successfully switches over 104 cycles with less than 5μA programming current. Extreme scaling of the device down to 6nmx6nm is realized by the CNT/AlOx/CNT cross-point structure and 104 switching cycles are achieved. This work is the first step toward RRAM with nm-scale electrodes. It paves the way for future high density, low power non-volatile RRAM memory application.