Inclusion Criteria:
* Diagnosed with Down Syndrome (DS), either with a karyotype or a compatible typical phenotype.
* Age over 40 years at time of screening.
* Symptomatic Alzheimer's Disease (AD) dementia, based on change in functionality and neuropsychological tests' results. Different cut-off points will be established to diagnose dementia depending on the level of intellectual disability of the individual, according to previous experience (Benejam et al; 2020): in adults with mild intellectual disability, a CAMCOG-DS score of 80 and an mCRT score of 29 will be chosen, whereas values of 56 and 28, respectively, will be used in subjects with moderate intellectual disability. Doubtful cases (e.g., with compromised functionality, but without alteration in the neuropsychological assessment) or those unable to complete the evaluation will be categorized by consensus among expert clinicians, using all available clinical information.
* Willing and able caregiver who has daily contact with the study subject.
* Subjects and caregivers must be able to comply with the prescribed regimen of study treatment throughout the course of the study and meet a minimum required time commitment of biannual in-person visits.
* Any concurrent treatment for AD approved by the European Medicines Agency (EMA) must be stable for at least 30 days prior to screening and at least 60 days prior to study day 1. Other medications (except for those listed under exclusion criteria) are allowed as long as the dose is stable for 30 days prior to screening.
* Subjects and/or their caregivers must be able to provide their consent before participating in any study-related procedures.
Exclusion Criteria:
* Cognitive changes attributable to causes other than AD (for example, but not limited to, uncorrected visual or hearing deficit, severe, untreated sleep apnea or uncontrolled thyroid disorders).
* Previous history of adult-onset epileptic seizures (over 18 years old).
* Treatment with any kind of antiepileptic drugs, benzodiazepines, narcotics.
* Significant comorbidities or analytical abnormalities, such as:
* Any unstable and/or clinically significant medical condition likely to hamper the evaluation of safety and/or efficacy of the study (eg, moderate and/or severe untreated obstructive sleep apnea, clinically significant reduction in serum B12 or folate levels, clinically significant abnormalities of thyroid function, stroke, or other cerebrovascular conditions), as per investigator's judgement.
* Severe renal dysfunction (creatinine clearance \< 30 mL/min), which would affect serum levetiracetam levels, or any other medical condition which is determined by the investigators to potentially create an undue risk for an adverse effect.
* Concomitant or past history psychiatric or neurologic disorder other than those considered to be related to AD (eg, head injury with loss of consciousness, symptomatic stroke, Parkinson's disease, severe carotid occlusive disease, transient ischemic attacks \[TIAs\]).
* Significant risk of suicide, defined using the C-SSRS as the subject answering "yes" to suicidal ideation questions 4 or 5 or answering "yes" to suicidal behavior within the past 12 months.
* Deviations from normal values for hematologic parameters, liver function tests, and other biochemical measures, judged to be clinically significant by the investigator.
* Participation in another clinical trial within 3 months of screening.
* Hypersensitivity to the active ingredient, other pyrrolidone derivatives, or any of the excipients
* Pregnant and breastfeeding patients
