We have found current-voltage (I-V) and conductance (dI/dV) characteristics of arrays of nanoscale tunnel junctions between normal metal electrodes to exhibit suitable features for primary thermometry. The current through a uniform array depends on the ratio of the thermal energy kBT and the electrostatic charging energy Ec of the islands between the junctions and is completely blocked by Coulomb repulsion at T = 0 and at small voltages eV/2 ≤ Ec. In the opposite limit, kBT ≫ Ec, the width of the conductance minimum scales linearly and universally with T and N, the number of tunnel junctions, and qualifies as a primary thermometer. The zero bias drop in the conductance is proportional to T-1 and can be used as a secondary thermometer. We will show with Monte Carlo simulations how background charge and nonuniformities of the array will affect the thermometer.