Background/Objectives: The main neurodegenerative diseases (NDs) - Alzheimer’s disease (AD), Parkinson’s disease (PD), multiple sclerosis (MS), and amyotrophic lateral sclerosis (ALS) - represent a growing global health burden with no available disease-modifying therapies. Curcumin, a polyphenol from Curcuma longa, is a promising candidate owing to its pleiotropic antioxidant, anti-inflammatory, and neuroprotective profile, but its oral bioavailability (<1%) and blood-brain barrier (BBB) penetration (<0.1%) severely limit its clinical application. The objective of this work was to critically examine the therapeutic potential of curcumin in NDs, focusing on advanced drug delivery systems (DDS) designed to overcome its pharmacokinetic barriers. Methods: Narrative, non-systematic review of PubMed/MEDLINE, Scopus, and Web of Science, structured around five areas: molecular mechanisms, pharmacokinetic barriers, evolution of DDS, disease-specific applications, and translational limitations. Results: Curcumin exhibits neuroprotective activity in preclinical models of the four NDs analysed, acting on six interconnected mechanisms and the gut-brain axis. Four generations of DDS have been developed, from phytosomes and clinically used lipid dispersions (Meriva®, BCM-95®, Longvida®, Theracurmin®) to fourth-generation systems (biomimetic nanoparticles, MOFs, microneedles, 3D scaffolds, hydrogels, carbon dots) that substantially increase bioavailability in preclinical studies. Combination strategies, such as curcumin with resveratrol and dutasteride, show preliminary clinical signals in ALS. However, clinical translation remains limited: over 80% of positive animal findings have not been replicated in humans, formulation characterization is frequently incomplete, and most trials lack CNS-exposure biomarkers. Conclusions: DDS are essential to unlock curcumin’s therapeutic potential in neurology, but its clinical translation requires phase II/III trials with well-characterized formulations, cerebrospinal fluid exposure biomarkers, and adaptive designs.