I have read several books that identify archaeological evidence in support of hundreds and hundreds of biblical places, cultures, and historical events. What is the archaeological evidence for the events in the Book of Mormon?
Depending on the scope of this question, it can be very difficult to give any satisfactory answer to. Archaeology primarily concerns itself with artifacts that have been left behind from ancient days, whereas the Book of Mormon deliberately avoids going into too much detail on the culture of its people and explaining details that might aid archaeological research, preferring to focus on more sacred matters. Also, it appears that their preferred method of construction was in wood, which doesn't lend itself well to the generation of archaeological ruins:
From Helaman chapter 3:
So we see that they considered timber to be the preferred method of construction. When it was unavailable, they developed new techniques for constructing houses and buildings of cement, and eventually became "exceedingly expert in the working of cement," but still placed a high value on wood as a construction material. (Having lived in South America, where many modern homes are built of cement and bricks, I can certainly understand this attitude. Cement is a rather good conductor of heat, which is an undesirable quality at low latitudes!)
The cement settlements are interesting. This was once derided as a hopeless anachronism in the Book of Mormon: the working of cement was an Old World art that never existed in pre-Colombian America! Right up until the mid-twentieth century, that is, when archaeologists started finding cement settlements in the vicinity of modern-day Mexico City, built with an extremely high degree of skill ("exceedingly expert") dating back to approximately the same time as the Book of Mormon states that these colonists moved in. (As reported in Concrete Evidence for the Book of Mormon.)
However, this settlement is an exception rather than a general rule. The people in question preferred construction in wood or simply living in tents, neither of which generates much in the way of ruins. Also, Book of Mormon archaeology is at a second disadvantage when compared to Biblical archaeology: on the Biblical side, we know exactly where to start looking! Modern-day Jerusalem is located on the side of the same mountain it's been on since at least the days of Melchizedek, to give just one example.
It will not be surprising, if you understand this, to hear that much of the truly interesting archaeological and historical evidence to arise in support of the Book of Mormon actually comes from discoveries in the Old World. To continue the earlier theme of Book of Mormon details once thought to be ridiculous, the book mentions two different men by the name of Alma. This was once held up as proof of Joseph Smith's hopeless inexperience: Alma is a Latin name, not a Jewish name, and a feminine Latin name at that!
...until the Dead Sea Scrolls were discovered, more than a century after the publication of the Book of Mormon, that is. One of the scrolls contains a reference to one "Alma, son of Judah." Oops.
Then we have the Lachish Letters, a group of clay tablets found in the ruins of an ancient Jewish city, dating back to just a few years after the start of the Book of Mormon. They describe several non-Biblical details of the local political climate which fit hand-in-hand with the corresponding Book of Mormon narrative. LDS scholar Hugh Nibley wrote a very detailed article on the letters, summarizing it at the end with 18 points of remarkable similarity between the two, and concluding that
(Two Shots in the Dark, first half)
Entire volumes have been written, and then, sadly, largely ignored, regarding the authenticity of the introductory chapters of the Book of Mormon, the ones that deal with travels in the Old World. To give just one example which is freely available online, see Lehi in the Desert, again by Dr. Nibley. He examines cultural details, religious and dream imagery, historical background, and even the names used, and shows that it is all remarkably consistent with the time and place it purports to originate from.
Stronger than any external evidence for or against the authenticity of the Book of Mormon, though, is internal evidence. There is a word for a literary work that claims a certain authorship but does not actually originate the way it claims to: forgery. And there are well-established tools and techniques for the detection of literary forgery, and they do not require the support of external evidence. But strangely enough, these formal methods don't tend to get applied in adversarial analyses of the Book of Mormon.
On the surface of it, there's no good reason not to; the Book of Mormon is a forger's nightmare! A fairly large work, claiming to be historical in nature, disseminated as widely as possible and inviting all to read and critically examine it? What is the forger thinking?!? This violates every rule of producing a successful forgery! But the two most important principles in the detection of literary forger tend to be consistently ignored:
When the formal, scientific techniques for detecting forgery are applied to the Book of Mormon, instead of emotionally-driven attempts to reach a predetermined conclusion, the book is shown to be genuine. (Dr. Hugh Nibley, New Approaches to Book of Mormon Study)
I could go on, (and on and on and on; as I said before, there have been entire volumes written on the subject,) but this answer is getting to be too long as it is. Suffice it to say that a large corpus of evidence exists which, while not enough to conclusively prove the truthfulness of the Book of Mormon, (they haven't even managed to do that with the Bible, so it would be unfair to hold another work that claims to be scripture to such a high standard,) definitely demonstrates that the book is worthy of serious attention and consideration.
There is a large and growing body of work by LDS scholars offering archaeological and anthropological evidence in support of the Book of Mormon: an introduction to it may be found on the FairMormon Wiki, However, it is not expected that such evidence will ever be conclusive.