Ashers Baking Company's hearing is taking place in Belfast High Court, with Northern Ireland's Lord Chief Justice, Sir Declan Morgan, and two other senior judges presiding over it.
It's after the Company, run by Daniel and Amy McArthur, told customer Gareth Lee they were unable to process his order for a cake because of the message he requested to be put on it.
Mr Lee asked for a cake with the Sesame Street characters Bert and Ernie, with the words 'Support Gay Marriage' above them.
The Equality Commission of Northern Ireland then brought a case against Ashers Baking Company, and in March 2015, District Judge Isobel Brownlie found Ashers directly discriminated against Mr Lee.
The couple were ordered to pay damages of £500, but subsequently appealed the discrimination verdict.
Speaking outside Belfast High Court, Daniel McArthur said: "We were simply unwilling to endorse a campaign for a new law that so clearly goes against what the Bible says about marriage. And for that we were punished.
"Christians are law-abiding citizens and we expect the law to protect us as much as anyone else.
"We hope that the judicial system will now make the correct decision and protect our freedom to carry out our work without being forced to violate our consciences.
"As a family we have found the whole legal process very difficult.
"We would rather not have to be here today. But we knew that we had to appeal, not only on our own behalf, but on behalf of other family businesses who could be forced to endorse or promote views with which they profoundly disagree.
"We appeal to those who would condemn us for our actions to consider what they would have done if they were required by law to use their creative abilities to help promote a cause which went against their strongest convictions.
The appeal is expected to be heard both Wednesday and Thursday and is being watched intently by both Christian and non-religious observers.
If Ashers' appeal is rejected, it is possible other businesses will legally have to forego deeply held beliefs in the service of clients.
Some have suggested it could lead to a Muslim-owned printer having to publish a cartoon of the Prophet Muhammad, or an atheist web designer may have to make a website stating that the world was made by God in seven literal days.
