Yes, as per Ryan's comment you can just take an infinite direct sum of finite groups. However the more interesting problem is: are there (infinite) $\textit{finitely generated}$ groups with all elements of finite order. ?

The answer to this was open for a long time, but it is indeed yes. In fact this was known as Burnside's problem

The first examples were given by Golod & Shafarevich.

Yes, as per Ryan's comment you can just take an infinite direct sum of finite groups. However the more interesting problem is are there $\textit{finitely generated}$ groups with all elements of finite order.