With both Manchester City and Liverpool in the semi-finals of their respective European competitions, City's place in the top 4 precarious and Liverpool's a non-existent possibility, there has been some concern and confusion over what Arsenal need to do to finish in the Champions League places. Part of this confusion is down to UEFA's new rule, whereby the winner of the Europa League qualifies automatically for the Champions League group-stage, if they hadn't already.
Arsenal, though, needn't worry--as long as Arsenal finish in the top 3. There is, however, a scenario where Arsenal could finish in the top 4 and not qualify for the Champions League. If Liverpool win the Europa League, Manchester City's league form continues to nosedive leading to City finishing outside the top 4, but, like Chelsea in 2012, win the Champions League, and Arsenal's league form continues it's decline and the Gunners end up finishing 4th, Arsenal would be in the group stage of the Europa League, not the Champions League. This handy guide, from Oliver Kay, via the Premier League press office, details every possibility.
Unlike 2012, a league now gets an additional place in the Champions League if the winner of a European competition has not already qualified for the competition. In that eventuality, if Arsenal finish 3rd, Arsenal would play a Champions League play-off, rather than go directly into the group stage. Thus, all Arsenal need to do is win their own matches. Given recent results, though, that's easier said than done.