A Lutheran church and Reykjavik's most prominent landmark, Hallgrímskirkja is a colossal structure whose main tower stands a whopping 73 meters tall. It was designed by renowned Icelandic architect Guðjón Samúelsson, with an external appearance that pays homage to the country's unique geologic makeup (anyone who's seen the cooled columns of basalt that dot the Icelandic landscape will know what we're talking about). Inside its beautiful, white, sparsely decorated interior you'll find plenty of cavern-esque tranquility to enjoy, as well as a spectacular 5,275 pipe organ by none other than Johannes Klais himself. Good stuff.
If awe-inspiring architecture and historic instruments aren't enough for you, Hallgrímskirkja has a couple other tricks up its sleeve, including the opportunity to ascend the tower via lift and enjoy a panoramic view over southwest Iceland for about $5 USD. Then there's also the prominent statue of Leif Erikson out front, a gift from the U.S. in 1930 to commemorate the founding of Iceland's parliament and pay homage to one of Europe's more under-appreciated explorers. All in all, if you're spending any time at all in the capital, this is definitely a building you should spend at least 10 minutes checking out.