Arsène Lupin is perhaps the most famous thief in the world of literature, responsible for daring heists and even more daring escapes from the law. This "Gentleman Burglar" starred in a long-running series of novels and short stories written by Maurice Leblanc from 1905 until about 1941. He has since gone on to reappear in various other media ever since.
And due to his popularity, has had a few run-in (unofficially/mostly non-canon) with Sherlock Holmes, the first of which might be one of the earliest forms of 'crossover'.
The First Meeting
Called in to solve an ancient riddle, (as well as help apprehend Lupin) Holmes unknowingly bumps into Lupin (who's in disguise) on his way to meet the authorities. As can be expected, Holmes solves said riddle in relatively short order, only to find out (upon figuring out the last part of the puzzle) that Lupin has beat him to the punch and got away with various valuables. However, having anticipated that the riddle would be easily solved by the great detective, Lupin, (no slouch himself in the brains department) as a mark of respect and admiration, has left his car waiting for Holmes at the exit of the secret tunnels in question... as well as his pocketwatch that the thief nabbed from him.
While he came out on top in their first go, Lupin realises that despite their brief meeting, Holmes' amazing deductive abilities would prevent him from fooling the detective (via disguise) a second time.
As one can imagine, Sir Arthur Conan Doyle didn't exactly take too kindly to this use of his character that subsequent meetings between the two necessitated the use of more lawyer-friendly names to skirt around the concept of copyright. So some versions of the above tale (and subsequent meetings) have it as 'Herlock Sholmes' (or Hemlock Shears).
The two rivals meet again in 1906's "The Blonde Lady" (or Blonde Phantom in some versions), that has Holmes again called back to the continent to help French authorities reclaim the priceless 'Blue Diamond', which Lupin has stolen thanks to a blonde female accomplice. What proceeds next is an ongoing battle of one-upmanship between the two evenly-matched opponents that ends with Holmes getting the upper hand, delivering the Gentleman Burglar into the hands of the police. Yet despite this, Lupin manages to escape in time to wave Holmes off and bid him farewell at a train station.
Next came 1907's "The Jewish Lamp" which has Holmes receiving yet another appeal from France to come help recover a Jewish lamp of some significant value. Shortly afterwards, he receives yet ANOTHER letter from France, from Lupin himself, begging the Master Detective (in not so specific terms) not to intervene. Despite this attempt to dissuade his rival, Holmes accepts the case. After yet another harrowing case, (ending with Lupin and Holmes in a battle of wits and nerves as they sit in a sinking boat) Holmes discovers that if he follows through with the arrest of the Burglar, an innocent woman will be dragged into the ensuing mess and wrongfully jailed.
The next and final meeting of the two (of the Leblanc stories) is 1909's epic novel "The Hollow Needle" where Holmes plays a pivotal, yet background role, (the main character who we see is a young man named Isidore Beautrelet) till the end when he arrives (once again) at one of Lupin's special hideouts (the aforementioned Hollow Needle), having solved a great deal of the case in seclusion (behind the scenes) with the proper authorities at hand. This shatters Lupin's plans to escape with a massive amount of treasure and retire from crime with a woman he has just married. A vicious battle between the two ensues that ends with the accidental death of Lupin's lover.
Lupin has crossed swords with Holmes off and on since then, in various media.
Frogwares' "Sherlock Holmes versus Arsene Lupin" (aka: "Sherlock Holmes: Nemesis"), a 2007 computer game that (in a more official capacity) pits the two against each other in the 'Frogwares Holmes' Canon, where the Gentleman Burglar threatens to steal various items of historical value from London unless Holmes can stop him. But as one can guess, there's much more than meets the eye to this.
Black Coat Press has had many crossovers between these two titans of crime literature in their continuing anthology series "Tales of the Shadowmen", as well as other characters of the world of pulp literature. These stories recognize the Leblanc novels as in their continuity (as well as the 'Wold Newton Family' concept).