After his initial presentation at the Santiago Bernabeu, the former Liverpool fullback took questions from the local press.
Alexander-Arnold began: "It's very exciting. It's a day I've been waiting for a long time. I'm proud to be here; it's a dream come true. I can't wait to get started.
"I've been waiting a long time. A couple of weeks, not years. I'm very excited to be here. You have to respect the club. I grew up watching Madrid win back-to-back Champions League titles. It doesn't surprise me to see Madrid winning titles. I want to be part of all the success from now on. The club has a winning mentality. We're going to the United States to win the Club World Cup."
Alexander-Arnold says he's already had a good conversation with coach Xabi Alonso, a former Liverpool star, who also has only just arrived.
He continued: "It's good to be able to talk to him. I've told him he's an idol of mine, and working with him is a dream come true. He's impacted the way I play. He did impossible things and has influenced me in terms of working and training hard. We haven't even met yet, and I'm sure we'll have a chat in the next few days about what he expects from me.
"He's a new coach, and it might take a while to get everyone to understand that. But he's already shown what he's all about. I'm going to try to absorb as much as I can."
Bellingham talked to me about Real Madrid
On Jude Bellingham, he said: "We've talked. I don't think it's what people thought. We've talked about Liverpool and Real Madrid. I wanted to know what Real Madrid is like. Many people think he played an important role in getting here, but Real Madrid needs no introduction.
"To be honest, it's not something you can dream of. It's beyond the realm of possibility for many players; there are only a handful of English players who can do it. When I was younger, I didn't think I could be where I am today. I was focused on learning. Today is a dream for me."
On his departure from Anfield, which drew an initial backlash from Reds fans, Alexander-Arnold insists there's no grudge.
He said: "Everything I've achieved with Liverpool has been incredible, it'll never be forgotten. But I've come out on a good note, I'm happy with the way I've done. I couldn't say anything bad about the fans. They've all been incredible, they've supported me so much. It's been an incredible help. The owners have thanked me for everything I've done for the club and that I'm always welcome. To be told all that is incredible. Now being here is very exciting for me. It's unbelievable. Something not many players get to experience.
"I've been fortunate enough to play for the biggest team in England and now the biggest in Spain. Both have dominated and won their leagues. It's an honor to be able to represent both. They have many similarities. Liverpool expected us to win titles, and they do here too."
Why 'Trent'
Alexander-Arnold will wear 'Trent' as his name on the back of his shirt.
He explained: "It's easy. Whenever I travel around Europe, people find the name difficult. I thought the easiest thing to do was to simplify it."
Asked if he'll be more appreciated in Spain than in England, Alexander-Arnold added: "I haven't given it much thought. I do what the coaches I've played for tell me. Each one asks different things. I do what they ask of me in the best way possible. I'm concerned with helping my teammates win titles and give my best."