The {\em bandwidth problem} is the problem of enumerating the vertices of a given graph $G$ such that the maximum difference between the numbers of adjacent vertices is {\em minimal}. The problem has a long history and a number of applications and is known to be $NP$-hard, Papadimitriou \cite{Pap76}. There is not much known though on approximation hardness of this problem. In this paper we show, that there are no efficient polynomial time approximation schemes for the bandwidth problem under some plausible assumptions. Furthermore we show that there are no polynomial time approximation algorithms with an absolute error guarantee of $n^{1-\epsilon}$ for any $\epsilon >0$ unless $P=NP$.