  We describe some new general constructions of $p$-adic $L$-functions attached
to certain arithmetically defined complex $L$-functions coming from motives
over $\bold Q$ with coefficiens in a number field $T$, with $[T:\bold
Q]<\infty$. These constructions are equivalent to proving some generalized
Kummer congruences for critical special values of these complex $L$-functions.
