p-adic L-function: survey + epsilon

To a given motive or a Galois representation over a number field, we can associate a Hasse-Weil L-function, which is conjecturally a meromorphic function defined over the whole complex plane. According to the Langlands conjecture this Hasse-Weil L-function corresponds to the automorphic L-function of the corresponding automorphic representation.
If we take a prime number p, which is an ordinary prime of this motive, we can consider the p-adic counterpart of this Hasse-Weil L-function, which is called the p-adic L-function. The existence of such a p-adic L-function is conjectural, but for motives of lower ranks the construction is known. In this talk, we can not give an updated overview of the known cases and recent works, but we recall the classical conjecture of the existence (mainly due to Coates and Perrin-Riou) and we explain what we have to check to construct the p-adic L-function.
If time permits, we want to discuss a very short overview of future perspectives on the p-adic L-function associated with (a p-adic deformation of) a motive.