Cypriot citizenship can be obtained via naturalization based on one’s years of residence on the island. More specifically: 

• Third Country Nationals can be naturalized as Cypriots once they have completed 7 years of legal residence on the island. Highly skilled individuals working in companies in Cyprus in the fields of tech, shipping and pharmaceutics can apply after 4 or 5 years of residence depending on the level of knowledge of the Greek language, i.e. A2 or B1 respectively. Official guidelines can be found here. 
• EU Nationals are allowed to apply for Cypriot citizenship after five years of residence on the island;
  In addition to the minimum time spent, one must also fulfil the required conditions mentioned above.
• Cyprus Citizenship via Spousal Privilege: Cypriot citizenship may also be obtained as the spouse of a Cypriot citizen under the following parameters: 
  Foreign spouses (EU Nationals and TCNs) of Cypriot Nationals who have been married for 3 years and have completed 2 years of residence in Cyprus, prior to the date of the application are allowed to apply;
  Foreign spouses (EU Nationals and TCNs) of overseas Cypriot Nationals who have completed 3 years of marriage can submit the said application at a Cyprus Consulate or Embassy overseas.
  EU Nationals: not sure about the below
  European Union citizens who wish to take up residence in Cyprus need to obtain an EU Registration Certificate within 4 months from the last entry to Cyprus.