  In 1950 G. Giuga studied the congruence $\sum_{j=1}^{n-1} j^{n-1} \equiv -1$
(mod $n$) and conjectured that it was only satisfied by prime numbers. In this
work we generalize Giuga's ideas considering, for each $k \in \mathbb{N}$, the
congruence $\sum_{j=1}^{n-1} j^{k(n-1)} \equiv -1$ (mod $n$). It particular, it
is proved that a pair $(n,k)\in \mathbb{N}^2$ (with composite $n$) satisfies
the congruence if and only if $n$ is a Giuga Number and $
\lambda(n)/\gcd(\lambda(n),n-1)$ divides $k$. In passing, we establish some new
characterizations of Giuga Numbers.
