In the countries of Western Europe in the 1990s increased the number of persons seeking international protection. States are confronted with major refugee crises and intensively develop cooperation in the field of asylum and migration. Asylum is considered to be the protection provided by the state on its territory to a person who has fled from his state to endanger life or serious human rights violations. The term derives from the Greek polis and means inviolability or shelter, refuge or a hiding place. A place that provides security and protection from persecution and danger in general. Asylum represents every shelter in the face of persecution, inconvenience, illness, old age, often in the form of an institution (a refuge for uninhabited children, the elderly, the poor and the homeless). The right to asylum is one of the fundamental human rights. In Croatia the right to asylum is guaranteed by the provision of Art. 33st 1st. Croatian Constitution, according to which: "Foreign citizens and stateless persons may obtain asylum in Croatia, unless they are prosecuted for non-political crimes and activities contrary to the basic principles of international law." Access to the asylum system is a basic prerequisite for the realization of asylum rights in practice so that the state can identify persons who need international protection amongst other migrants during the contemporary migration movements, it is necessary to ensure access to the territory and the asylum system. Every legal and business person is entitled to apply for asylum in his / her own name. Adult family members may be covered by the requirement of a member of the family, provided that they agree. Asylum seekers have the right to health care, which includes urgent medical assistance and the need to treat the disease. The paper analyzes the concept of asylum, the process of acceptance and access to asylum, asylum seekers' rights and obligations and the role of the Croatian Red Cross.