This study will investigate the safety, tolerability and blood pharmacodynamics of treatment with oral administration of AZD7325 at 5 mg BID, 15 mg BID, and placebo BID, in adults with Fragile X Syndrome. The study also will also investigate measures of efficacy and biomarkers during treatment.

Inclusion Criteria:

• Diagnostic confirmation of full mutation FXS
• 50 ≥ Age ≥18 years. Males and Females included in study.
• General good health as determined by physical exam, medical history and laboratory work up.
• FXS genetic reports at screening
• IQ less than or equal to 80. Note: IQ cutoff is used as a means to exclude cases of females with FXS who have the full mutation, but may have neurotypical development (ie: do not have the full FXS phenotype despite positive FXS genetic testing) due to variability in X chromosome inactivation patterns.
• Male study participants who are sexually active with a female partner of childbearing potential must be surgically sterilized, practicing abstinence, or agree to use highly effective methods of birth control (defined in the list below), and not rely on barrier methods and spermicide alone, from the time of screening until 1 week after final dose of study drug. Male study participants must also not donate sperm from the time of screening until 1 week after final dose of study drug. Given that AZD7325 is not mutagenic, there is no mandatory requirement for condom use, either for avoidance of procreation or in the case of treated males with a pregnant partner.
• Women of childbearing potential may be included in the study provided they are established on, and continue to use, highly effective contraceptive methods from the time of screening until 1 week after the final dose of study drug. Highly effective methods of contraception associated with inhibition of ovulation (either oral, intravaginal or transdermal), progestin-only hormonal contraception associated with inhibition of ovulation (either oral [specifically Micronor, Nor-QD or their generic equivalents], injectable or implantable).
• Aberrant Behavior Checklist total score of 20 or higher at screening

Exclusion Criteria:

• Concomitant use of modulators of GABA A neurotransmission. (examples)
• Use of more than three psychotropic drugs that do not directly impact GABA transmission, and/or unstable dosing of any psychotropic medication in the 4 weeks prior to baseline visit.
• Subjects are prohibited from use of strong and moderate modulators of CYP3A and CYP2C19 during the screening (at least 2 weeks before initiation of the study) and treatment periods of the study. Such prohibited drugs are outlined in http://www.fda.gov/downloads/drugs/guidancecomplianceregulatoryinformation/guidances/ucm292362.pdf
• CNS-suppressing agents such as central analgesics, muscle relaxants, benzodiazepines, other sedatives, and should also limit alcohol intake to ≤1 alcoholic beverage per day.
• Unstable seizure disorder as defined by any seizure in the 6 months prior to baseline visit and/or a change in any anti-convulsant drug dosing in the 60 days prior to study entry.
• All patients with abnormal baseline safety lab assessments including, but not limited to ALT or AST greater than 1.5 the upper limit of normal, total bilirubin or creatinine greater than 1 time the upper limit of normal or other clinically relevant lab abnormality or abnormality in ECG, HR or BP at screening as judged by the investigator.
• Clinical relevant history or presence of any medical disorder judged by the investigator at potentially interfering with this trial.
• History of or current abuse of drugs or alcohol including prescription medication.
• For female subjects of child bearing potential (women 50 & under is "amenorrhoeic for 12 months or more (following cessation of exogenous hormonal treatments - if these have been previously taken) and with luteinizing hormone (LH) and follicle stimulating hormone (FSH) levels in the post-menopausal range) a positive pregnancy test.