The characters for Reimu's last name "博麗" (Romanized: Hakurei) literally mean something like "esteemed companion", the first word "博" (Haku) means "wide" or "exposition" while "麗" (Rei) means "lovely", "graceful", and "beautiful". Her first name "霊夢" (Romanized: Reimu) means "revelatory dream" or "a vision", the first word "霊" read as "Rei" means "spirit" or "soul" while "夢" (Romanized: Yume) is read "Mu" and means "dream".
The surname Hakurei is also another name for sake, as the maker of the Touhou series, ZUN, is quite the connoisseur of "spirits", so much so that fans often speculate over whether ZUN might be or has been an alcoholic.
Reimu has appeared as a playable character in every integer-numbered game in the Touhou series.
Reimu makes an official appearance as "Flying Maiden" in Graffiti Kingdom, as ZUN was involved in the production of it.
Reimu appeared on the cover of all the PC-98Touhou games, from the first game in the series to the fifth.
Reimu was never seen flying with her own power until Embodiment of Scarlet Devil. In the previous games, if Reimu needed to go somewhere far, she would ride on her turtle familiar, Genjii.
Although her many profiles suggest that the "Hakurei Clan" and the shrine Reimu lives at have been in Gensokyo for long enough to be considered ancient, there has never been any mention of Reimu's family members, ancestors, etc. It is originally stated that she lives alone at the shrine, but other rarely seen characters (Mima and Shingyoku) are known to stay there or near there as well.
Two or more Reimu(s) may exist. The circumstantial evidence suggests 靈夢 (from the PC-98 games) and 霊夢 (from the Windows games) may be different people. However, this is extraordinarily unlikely, as both Yuuka and Alice recall Reimu, and ZUN uses the new kanji to refer to the old Reimu now. Also, her bio in Embodiment of Scarlet Devil's "おまけ.txt" file says, "Chronologically this title is not too far from the previous title, thus her age stays the same as before". Since the previous title in question would be Mystic Square, the last game in which PC-98 Reimu appeared in, and Embodiment of Scarlet Devil is the first game in which Windows Reimu appears, the bio appears to further debunk this theory.
Some people state that the PC-98 Reimu may have been the current Reimu's ancestor, and that Yuuka and Alice seem to compare her instead of recalling. In the Touhou Kinema-Kan animations, a different-looking Reimu engages in battle with Yukari after setting up the border, thus summing it up along with the fact that Reimu's family may have also known Yukari for a long time.
In Imperishable Night Reimu is the only one whose partner doesn't mention the danger of the full moon.
When playing Yukari's story mode in Scarlet Weather Rhapsody, Reimu breaks down in tears in the ending after Yukari defeats Tenshi and reduces the Hakurei Shrine to rubble for the second time, with Sakuya and Youmu looking on helplessly (although Suika offers to help rebuild it). This marks the first and only time where she is seen crying in a storyline scenario.
Reimu's Last Word 「夢想天生」 "Fantasy Heaven" (Pronounced: Musou Tensei) is a homonym of 「無想転生」 "Unconscious Transmigration of Souls" (Also pronounced: Musou Tensei), Kenshiro's ultimate technique in Hokuto no Ken. In 12.3, if you activate Fantasy Heaven successfully in a 3rd round, a remix of Mystical Oriental Love Consultation in the style of the Fatal KO theme from the Hokuto no Ken fighting game will play. Fitting, because activation of the said Spell Card means the opponent is certainly already dead! Because of this it is now known as the Hakurei Hyakuretsu Ken to some fans; also, Kenshiro's catch phrase ("you are already dead") was also adopted to Reimu ("you have already donated").
In Touhou Hisoutensoku, one of her alternative palettes looks similar to VIVIT, the heroine of Seihou. Also, one of her palettes seems to be another maid homage, Ruukoto, or possibly Sanae's colors. The palette also seems to resemble Mima from Highly Responsive to Prayers. Her white/light gray palette seems to resemble Toki from Hokuto no Ken, and possibly similar to the White Reimu from Concealed the Conclusion. And finally, her white/light blue palette looks like a tribute to Nanoha, as they both tend to "befriend" their opponents with superior firepower.
In Lotus Land Story, when you clear Easy Mode with Reimu and just before the Name Registry screen shows up, you get a Congratulation screen with a drawing of her wearing an outfit that very closely resembles her outfit from Embodiment of Scarlet Devil (save for non-detached sleeves).
In Immaterial and Missing Power, Reimu appears with blue eyes, but in the other Windows games, she is seen with brown eyes. Also, the Image Profiles show her with black hair, but in the shooting games, she has brown hair. (This includes SWR and Hisoutensoku images)
Interestlingly, her Divine Arts "Wind God Kick" Spell Card resembles Chris' Twister Drive Desperation Move in The King of Fighters.
Reimu seems to be scared before and after the boss battle with the Five Magic Stones in Story of Eastern Wonderland, making this the only ever time that Reimu was scared and/or afraid of a boss before and after the battle.