A permutation pi=(pi_{1},...,pi_{n}) is consecutive 123-avoiding if there is no index i such that pi_i < pi_{i+1} < pi_{i+2}. Similarly, a permutation pi is cyclically consecutive 123-avoiding if the indicies are viewed modulo n. These two definitions extend to (cyclically) consecutive S-avoiding permutations, where S is some collection of permutations on m+1 elements. We determine the asymptotic behavior for the number of consecutive 123- avoiding permutations. In fact, we give an asymptotic expansion for this number. Strangely enough we obtain an exact expression for the number of cyclically consecutive 123-avoiding permutations. A few results will be stated about the general case of (cyclically) consecutive S-avoiding permutations. Part of these results are joint work with Sergey Kitaev and Peter Perry.