In this work, nanocomposites based on the triblock copolymer polystyrene-b-polyisoprene-b-polystyrene (SIS) thermoplastic elastomer filled with multiwall carbon nanotubes (MWCNT) were obtained by melt mixing. The nanocomposites were characterized by oscillatory rheometry, electrical resistivity, small angle x-ray scattering (SAXS) and transmission electron microscopy (TEM). The results showed that both, rheological and electrical percolation were achieved at MWCNT loadings between 1-3 vol.%. Rheological tests revealed that the insertion of MWCNT into SIS significantly enhanced the process of relaxation of SIS blocks. Resistivity measurements revealed that conductive nanocomposites were obtained at MWCNT loadings ~1.6 vol.%. The electrical resistivity decreased eleven orders of magnitude from neat SIS to SIS/ 5 vol.% MWCNT. Finally, SAXS and TEM showed that the melt mixing process and the presence of MWCNT hampered the self-assembly of SIS into well-ordered domains.