Was Professor Moriarty ever apprehended or killed by Sherlock Holmes?

Question

Professor Moriarty was Sherlock Holmes' arch enemy.

Was Professor Moriarty ever arrested or killed by Sherlock Holmes?

Answer 1

This is the plot of the story "The Final Problem".

"The Final Problem" is the story where Moriarty is introduced, as the most fiendish criminal in all of London, and also the story where he meets his end, apparently (but, as later revealed, not really) together with Holmes himself.

This short story begins with Holmes telling Watson all about his entanglement with Moriarty, how we wove a net around him for months until the evil professor was at risk of losing his liberty (i.e. arrest and imprisonment):

“‘You crossed my path on the 4th of January,’ said [Moriarty]. ‘On the 23rd you incommoded me; by the middle of February I was seriously inconvenienced by you; at the end of March I was absolutely hampered in my plans; and now, at the close of April, I find myself placed in such a position through your continual persecution that I am in positive danger of losing my liberty. The situation is becoming an impossible one.’

Holmes's plan is set to culminate on the following Monday, at which date Moriarty and all his associates and henchmen would be arrested. Since they are making constant assassination attempts on him, he escapes to the Continent together with Watson. The news on that fateful Monday shows that Moriarty himself escaped:

We made our way to Brussels that night and spent two days there, moving on upon the third day as far as Strasburg. On the Monday morning Holmes had telegraphed to the London police, and in the evening we found a reply waiting for us at our hotel. Holmes tore it open, and then with a bitter curse hurled it into the grate.

“I might have known it!” he groaned. “He has escaped!”

“Moriarty?”

“They have secured the whole gang with the exception of him. He has given them the slip. Of course, when I had left the country there was no one to cope with him. But I did think that I had put the game in their hands. I think that you had better return to England, Watson.”

“Why?”

“Because you will find me a dangerous companion now. This man’s occupation is gone. He is lost if he returns to London. If I read his character right he will devote his whole energies to revenging himself upon me. He said as much in our short interview, and I fancy that he meant it. I should certainly recommend you to return to your practice.”

Moriarty then personally follows Holmes and Watson in order to seek his revenge for the collapse and destruction of his criminal empire. The story ends when, having got Watson out of the way by a ruse, Moriarty comes face to face with Holmes and they engage in a final physical fight. Holmes's letter reveals his expectation of death, and Watson deduces that Holmes and Moriarty died together at the Reichenbach Falls:

“I write these few lines through the courtesy of Mr. Moriarty, who awaits my convenience for the final discussion of those questions which lie between us. He has been giving me a sketch of the methods by which he avoided the English police and kept himself informed of our movements. They certainly confirm the very high opinion which I had formed of his abilities. I am pleased to think that I shall be able to free society from any further effects of his presence, though I fear that it is at a cost which will give pain to my friends, and especially, my dear Watson, to you. I have already explained to you, however, that my career had in any case reached its crisis, and that no possible conclusion to it could be more congenial to me than this.

Doyle intended this to be the end of Sherlock Holmes, but public outcry at the popular detective's death was such that, some years later, Doyle and (in-universe) Watson took up their pens again to recount The Return of Sherlock Holmes, in which the first story, "The Adventure of the Empty House", reveals that Holmes didn't die but Moriarty did:

“This is, indeed, like the old days. We shall have time for a mouthful of dinner before we need go. Well, then, about that chasm. I had no serious difficulty in getting out of it, for the very simple reason that I never was in it.”

“You never were in it?”

“No, Watson, I never was in it. My note to you was absolutely genuine. I had little doubt that I had come to the end of my career when I perceived the somewhat sinister figure of the late Professor Moriarty standing upon the narrow pathway which led to safety. I read an inexorable purpose in his grey eyes. I exchanged some remarks with him, therefore, and obtained his courteous permission to write the short note which you afterwards received. I left it with my cigarette-box and my stick, and I walked along the pathway, Moriarty still at my heels. When I reached the end I stood at bay. He drew no weapon, but he rushed at me and threw his long arms around me. He knew that his own game was up, and was only anxious to revenge himself upon me. We tottered together upon the brink of the fall. I have some knowledge, however, of baritsu, or the Japanese system of wrestling, which has more than once been very useful to me. I slipped through his grip, and he with a horrible scream kicked madly for a few seconds, and clawed the air with both his hands. But for all his efforts he could not get his balance, and over he went. With my face over the brink, I saw him fall for a long way. Then he struck a rock, bounded off, and splashed into the water.”

TL;DR: yes. Holmes gathered enough evidence to convict Moriarty; he escaped arrest but Holmes killed him at the Reichenbach Falls.

Further reading:

• Why did Sir Arthur Conan Doyle decide to kill off this character?
• Is Professor Moriarty really Sherlock Holmes' arch-enemy in the way some people think?