In South Korea, international students who have completed their studies have several options for obtaining a work permit, permanent residency, and eventually citizenship. Here are some common paths:

1. Employment Visa (E-7): If you find a job offer in South Korea after completing your studies, your prospective employer can sponsor an Employment Visa (E-7) on your behalf. The E-7 visa is designed for individuals with specialized skills or knowledge. The specific requirements and eligibility criteria depend on the job position, qualifications, and other factors determined by the South Korean authorities.

2. Points-Based System: South Korea operates a points-based system called the F-2-7 visa, which allows foreign nationals to obtain permanent residency based on their qualifications, language proficiency, work experience, income, and other factors. By meeting certain point thresholds, you can apply for permanent residency, which offers long-term settlement in South Korea.

3. Marriage to a South Korean Citizen: If you are married to a South Korean citizen, you may be eligible to apply for a spouse visa (F-6) or a permanent residency visa (F-5). These visas allow you to live and work in South Korea. The specific requirements and procedures may vary, so it's advisable to consult the South Korean immigration authorities for detailed information.

4. Investment Visa: If you wish to invest in a business in South Korea, you can explore the option of obtaining an investment visa. The specific requirements and investment thresholds depend on the type and scale of the business. Successful applicants can obtain a visa that allows them to work and reside in South Korea.

5. Korean Citizenship: Obtaining South Korean citizenship typically requires residing in South Korea for a significant period (usually five years), meeting language proficiency and integration requirements, and fulfilling other criteria set by the South Korean government. The exact requirements may vary, so it's advisable to consult the South Korean immigration authorities for the most up-to-date information.