How does the valency of elements vary (i) in going down a group, and (ii) in going from left to right in a period of the periodic table ?

Valency is determined by the number of valence electrons required to complete an octet. In a group, the electronic configuration of the outermost shell is identical (e.g., Group 1 elements all have $1$ valence electron), so valency is constant. In a period, the number of valence electrons increases from $1$ to $8$. Elements with $1-4$ valence electrons lose or share them (valency $1-4$), while those with $5-7$ electrons gain or share electrons to reach $8$ (valency $8 - \text{valence electrons}$).