In this paper we study the sequences defined by the last and the last non-zero digits of $n^n$ in base b. For the sequence given by the last digits of $n^n$ in base b, we prove its periodicity using different techniques than those used by W. Sierpinski and R. Hampel. In the case of the sequence given by the last non-zero digits of $n^n$ in base b (which had been studied only for b = 10) we show the non-periodicity of the sequence when b is an odd prime power and when it is even and square-free. We also show that if $b=2^2{^s}$ the sequence is periodic and conjecture that this is the only such case.