You can filter and view conversations by the following by 

i) Channels you have enabled for your account. 

ii) Conversations that are assigned to you. (How to assign conversations?

iii) Conversations that you have marked to follow up. (How to mark conversations for followup?)

iv) Conversations that have been marked resolved. (How to resolve a conversation?)

To filter conversations, go to Live chat > and click on the filter icon on the top left. Next select on of the above mentioned filter options from the drop down.