Caffeine
Real vasoconstrictive and phosphodiesterase-inhibiting activity, a legitimate use for periorbital puffiness, and a much weaker case for everything else it is sold for.
What is caffeine?
Caffeine is a methylxanthine alkaloid, small (194 daltons) and moderately lipophilic, so it penetrates skin reasonably well — unusually favourable properties for a cosmetic active.
What it does, in order of evidential strength
Vasoconstriction is the best-supported effect and the basis for its use around the eyes: reducing local blood flow and fluid accumulation temporarily lessens the appearance of puffiness and, in some people, dark circles of vascular origin. The effect is genuine, immediate and short-lived.
Phosphodiesterase inhibition raises intracellular cyclic AMP, which promotes lipolysis. This is the mechanism behind body-contouring and cellulite claims. It is real in cell systems; the translation to visible change in human subcutaneous fat through topical application is much weaker, and any effect is temporary and largely attributable to local fluid shifts.
Antioxidant activity is present but unremarkable relative to dedicated antioxidants.
Hair density claims rest on caffeine's reported antagonism of testosterone-induced suppression of follicle growth in ex vivo systems. Interesting, and far from established clinically.
Levels
Studies of periorbital and contour effects generally use caffeine in the range of 1% to 5%. Vegalab uses it in Scalp Repair, Lumina Brightening Cream, the NanoDensity line and the Sculpting line. The Youlief ECW Ampoule lists caffeine at 0.0001% — 1 ppm — which is several orders of magnitude below the levels used in any of the work above, and the page should say so rather than imply activity.
What does caffeine not do?
It does not reduce body fat, remove cellulite or regrow hair. The puffiness effect is temporary and cosmetic. And caffeine applied to skin does not meaningfully contribute to systemic caffeine intake at cosmetic levels.