  Let $G$ be a finite group and exp$(G)$ = lcm$\{$ord$(g)$$\mid$$g \in G \}$. A
finite unordered sequence of terms from $G$, where repetition is allowed, is a
product-one sequence if its terms can be ordered such that their product equals
the identity element of $G$. We denote by $\mathsf s (G)$ (or $\mathsf E (G)$
respectively) the smallest integer $\ell$ such that every sequence of length at
least $\ell$ has a product-one subsequence of length $\exp (G)$ (or $|G|$
respectively). In this paper, we provide the exact values of $\mathsf s (G)$
and $\mathsf E (G)$ for Dihedral and Dicyclic groups and we provide explicit
characterizations of all sequences of length $\mathsf s (G) - 1$ (or $\mathsf E
(G) - 1$ respectively) having no product-one subsequence of length $\exp (G)$
(or $|G|$ respectively).
