Aris:
So what is the definition of Apostolic Succession and is the Catholic Church the only one that can claim it?
Dear Aris… just repeating something written before in answer to the second half of your question.
The oldest lineage of bishops which comes down to us in the 21st century who is a successor to St Peter is not actually the bishop (Pope) of the Church of Rome, but the bishop (Patriarch) of the Church of Antioch.
Check this article, from a Catholic source.
melkitecathedral.org/melkite/history3.htm
Peter’s First See
**The evolution of the Patriarchate of Antioch
**
The article features a fascinating photograph of three bishops with the apostolic succession of Saint Peter, photographed all together in Damascus in 2001.
Three successors of Peter
– Syriac Orthodox Patriarch Ignatius Zakka I,
- Pope John Paul II,
- Greek Orthodox Patriarch Ignatius IV
- gather in the Syriac Orthodox Cathedral of St. George in Damascus, May 2001. (photo: L’Osservatore Romano)
Peter founded the Church of Antioch in 34 AD, and he remained there for 5-7 years. Paul (and Barnabas) came to Antioch to see Peter there and it was in Antioch that the dispute between Peter and Paul flared up about whether converts had to be circumcised. In order to resolve this Peter and Paul took the dispute to James in Jerusalem and James called all the Apostles to a Council in Jerusalem to make a determination.
Early than this, Antioch had received a large number of Christian refugees who fled Jerusalem after the martyrdom of Saint Stephen the deacon, a period of martyrdom in Jerusalem which Paul himself had initiated while he was still the uncoverted Saul!
To succeed him in Antioch Saint Peter consecrated Euodius (Evodius) as bishop of that city. Euodius was succeeded as bishop in Antioch by the great Saint and holy martyr Ignatius who was himself consecrated by either Saint Peter or Saint Paul. The Patriarch of Antioch is the 170th successor of Saint Peter. Here is a complete list of his apostolic succession from the holy Apostle Peter
antiochian.org/Patriarchate/patriarchs.htm
So the Church of Antioch founded by Saint Peter is a little bit older than Rome, and like Rome it has an unbroken apostolic succession going back to Saint Peter.