I've read both The Silmarillion and The Lord of the Rings. Both take place in the 'west' of Arda, and Beleriand (where most of The Silmarillion takes place) is destroyed. So where is the Middle-earth of LotR? Does anything from The Silmarillion take place there? Can we point to places on the map that exist in both epics?
I was about to answer in depth, but then an image search showed me an existing answer online, though there's quite a bit of extraneous text there. I'll borrow the maps from that blog, originally taken from Karen Wynn Fonstad's The Atlas of Middle-earth:
This is a map of Beleriand, in the first age. Conveniently, the Blue Mountains are circled:
And now a map of Middle-earth in the third age:
These are the same Ered Luin, Blue mountains. Beleriand, before it sank into the sea, was westwards of where the Shire is in the Third Age.
To supplement Anver's answer, here is another map, sourced from History of Middle Earth 7: The Treason of Isengard, which contains an extended account of the first LotR map:
This was drawn by CT from one of JRRT's working maps so it may be considered an accurate representation.
Here, the island of Himling corresponds to the hill of Himring in Beleriand, and Tol Fuin with the highlands of Dorthonion/Taur-nu-Fuin. To quote from CT's note regarding the Unfinished Tales redrawing:
Himling was the earlier form of Himring (the great hill on which Maedhros son of Fëanor had his fortress in The Silmarillion), and though the fact is nowhere referred to it is clear that Himring's top rose above the waters that covered drowned Beleriand. Some way to the west of it was a larger island named Tol Fuin, which must be the highest part of Taur-nu-Fuin.
Although Christopher Tolkien notes that "the fact is nowhere referred to", it actually is; in HoME 7: The Treason of Isengard, Chapter 6 "Council of Elrond (1)" text of "a single sheet of manuscript found in isolation" reads (in part):
Taur na Fuin (earlier name) became an island ... (so also Himling)
CT confirms in a footnote that this "single sheet" was only discovered after he had written the Unfinished Tales note; hence his original statement (now known to be incorrect).
The river in Lindon may be the remnants of the Gelion, and what's especially interesting is that if you examine the Beleriand map in The Silmarillion, you'll see that the westward protrusions of Ered Luin there (in the area of Mount Rerir/Lake Helevorn) correspond in shape to the same area in the map above (to the east of Himling).
Also interesting is a hint of the location of Utumno, in Return of the King (note 25 to Appendix A, discussing "the Lossoth, the Snowmen of Forochel"):
These are a strange, unfriendly people, remnant of the Forodwaith, Men of far-off days, accustomed to the bitter colds of the realm of Morgoth. Indeed those colds linger still in that region, though they lie hardly more than a hundred leagues north of the Shire.
From this it may be inferred that the remnants of Utumno are under the Icebay of Forochel, giving you another correspondence between the LotR maps and the tales of The Silmarillion.
In The Silmarillion we also read that, after the Battle of the Powers, which resulted in the first Chaining of Melkor:
In that time the shape of Middle-earth was changed, and the Great Sea that sundered it from Aman grew wide and deep; and it broke in upon the coasts and made a deep gulf to the southward. Many lesser bays were made between the Great Gulf and Helcaraxë far in the north, where Middle-earth and Aman came nigh together. Of these the Bay of Balar was the chief...
This seems to indicate that the Great Gulf is the Bay of Belfalas, assuming that its size in the First Age was comparable to its size in the Third.
The Great March of the Eldar also passes through some well-known locations:
And it came to pass after many years of journeying in this manner that the Eldar took their course through a forest, and they came to a great river, wider than any they had yet seen; and beyond it were mountains whose sharp horns seemed to pierce the realm of the stars. This river, it is said, was even the river which was after called Anduin the Great, and was ever the frontier of the west-lands of Middle-earth. But the mountains were the Hithaeglir, the Towers of Mist upon the borders of Eriador; yet they were taller and more terrible in those days, and were reared by Melkor to hinder the riding of Oromë.
And also of the wanderings of the Nandor:
Little is known of the wanderings of the Nandor, whom he led away down Anduin: some, it is said, dwelt age-long in the woods of the Vale of the Great River, some came at last to its mouths and there dwelt by the Sea, and yet others passing by Ered Nimrais, the White Mountains, came north again and entered the wilderness of Eriador between Ered Luin and the far Mountains of Mist.
And finally, RotK Appendix A discusses the origins of the Barrow-downs and their relationship to the First Age:
It is said that the mounds of Tyrn Gorthad, as the Barrowdowns were called of old, are very ancient, and that many were built in the days of the old world of the First Age by the forefathers of the Edain, before they crossed the Blue Mountains into Beleriand, of which Lindon is all that now remains.
And to supplement their answers, a map from the LotR Fandom site:
The fully drawn bit is (the north-west corner of) Middle-earth in the Third Age, the faintly drawn part is Beleriand. Again you can see Tol Fuin and Tol Himring/Himling.
EDIT: Jimmy Shelter commented (quite rightly):
It's worth noting that the southern coast of the drowned lands in this map is conjectural (it can only ever be as Tolkien never mapped it) but consistent with "Many lesser bays were made between the Great Gulf and Helcaraxë far in the north" in The Silmarillion. The general assumption is that the "Great Gulf" is the Bay of Belfalas.