  We study the problem of fairly allocating a set of chores to a group of
agents. The existence of envy-free up to any item (EFX) allocations is a
long-standing open question for both goods and chores. We resolve this question
by providing a negative answer for the latter, presenting a simple construction
that admits no EFX solutions for allocating six items to three agents equipped
with superadditive cost functions, thus proving a separation result between
goods and bads. In fact, we uncover a deeper insight, showing that the instance
has unbounded approximation ratio. Moreover, we show that deciding whether an
EFX allocation exists is NP-complete. On the positive side, we establish the
existence of EFX allocations under general monotone cost functions when the
number of items is at most $n+2$. We then shift our attention to additive cost
functions. We employ a general framework in order to improve the approximation
guarantees in the well-studied case of three additive agents, and provide
several conditional approximation bounds that leverage ordinal information.
