  We consider the problem of guaranteeing maximin-share (MMS) when allocating a
set of indivisible items to a set of agents with fractionally subadditive (XOS)
valuations. For XOS valuations, it has been previously shown that for some
instances no allocation can guarantee a fraction better than $1/2$ of
maximin-share to all the agents. Also, a deterministic allocation exists that
guarantees $0.219225$ of the maximin-share of each agent. Our results involve
both deterministic and randomized allocations. On the deterministic side, we
improve the best approximation guarantee for fractionally subadditive
valuations to $3/13 = 0.230769$. We develop new ideas on allocating large items
in our allocation algorithm which might be of independent interest.
Furthermore, we investigate randomized algorithms and the Best-of-both-worlds
fairness guarantees. We propose a randomized allocation that is $1/4$-MMS
ex-ante and $1/8$-MMS ex-post for XOS valuations. Moreover, we prove an upper
bound of $3/4$ on the ex-ante guarantee for this class of valuations.
