Can foreigners buy property in Cyprus?

In short - yes, foreigners are able to purchase property in Cyprus. What’s more, it’s easier to do so here when compared with many European countries. But there is a certain procedure.EU citizens are able to purchase property in Cyprus with no restrictions.

1. Overview
Foreign nationals (citizens of non-EU Member States, including foreign companies and companies controlled by foreign nationals as defined by law) wishing to acquire immovable property in Cyprus must first obtain permission from the local District Administration.
To obtain permission, an application must be submitted to the District Administration of the district in which the property for sale is located, together with all the necessary supporting documents.
No fee is payable for permission to purchase, and applications take between 2 and 3 weeks to process.
2. Prerequisites
Foreign nationals or foreign couples can obtain permission to purchase:
a) A plot or land not divided into plots measuring up to 4,000 m² for the purpose of building a dwelling for owner occupation. Couples are granted joint permission. 
b) Up to two units which may be at different stages of development. The units may be two dwellings or one dwelling and a shop measuring up to 100m² or one dwelling and an office measuring up to 250m².
    The above restriction applies in total to couples.
3. Certificates
Application form must be with the following documents:
-A survey
-A copy of the title deed
-A copy of the planning consent (only if separate title deeds have not been issued for the apartment or dwelling for sale) For plots in the process of being partitioned, a copy of the partitioning permit
-A copy of the contract of sale, with the requisite stamp duty affixed
-Floor plans of the dwelling or apartment
-Proof of the applicants’ financial standing
-Copies of the applicants’ and their spouses’ passports
-The applicants’ marriage certificate with a certified translation (unless it is in Greek or English and only if the spouses do not have the same surname and the wife's passport does not mention her husband's surname)
-A copy of the applicants’ Cypriot residence and work permits
-If the applicant is a company or a shareholder of a company, the following must be submitted in addition to the above: certificates of registration, shareholders, directors and registered office for the company, the company's memorandum and articles of association and information on its activities and staff, together with their social insurance numbers and the first page of the company shareholders’ and their spouses’ passports.
Government Source here.