Arepas: Corn flour based pancakes, sometimes made with cheese or slightly salted.
Empanadas: The closest comparison would be pastries. These are popular all over South America, so generally each country/region has their own recipe. The filling usually consists of meat, potato, vegetables and rice wrapped in a corn flour crust.
Tamal: Usually eaten for breakfast. A mixture of meat, chicken, potato, vegetables and yellow corn wrapped in plantain leaves and then boiled. Should be accompanied by a large mug of hot chocolate.
Ajiaco: Traditional thick soup based on three kinds of potatoes, chicken, avocado, dairy cream, herbs, corn, among others. Typically from the altiplano region, and considered the city's official dish.
Pizza and burgers. OK, can we really call these traditional Bogotá meals? One could surmise they're from here, seeing their omnipresence. The city does both quite well, and you just need to do a 360° turn to find some.
Options are many for casual dining, unsurprisingly for a Latin American city of seven million people. Bogotanos love food from all over, so you'll find a good mix of Colombian food, as well as cheap food from North America (especially pizza and burgers!) and Asia. Note that the Chinese food is almost always Colombianized, which can be pretty good anyway, but is almost never the real deal. Sushi is likewise easy to find, but usually of below-average quality. The clear exception is (upscale) Wok, which has several locations, and for a North American-price will serve you top-notch sushi and other authentic East and Southeast Asian dishes.
For lunch, definitely try a corrientazo—a small eatery that is only open for lunch, serving people on their lunch break a delicious full meal, with soup, fresh squeezed fruit juice, a meat dish, several starch offerings, and usually additional fruit on the side. You'll know corrientazos by their well-advertised and extremely limited menu, which often consists of only one available main course! Best options are usually traditional ajiaco, bandeja paisa, or fish. Sometimes the advertisement is just "ALMUERZO" (lunch) on a cardboard sign. Prices are astoundingly low: around $3,500-10,000/meal.
Rotisserie chicken is usually not far away, often called chicken "broaster," and is just fabulous. They'll pass you plastic gloves to wear while you eat it to keep your fingers clean.
If constant meat and starch isn't your thing, the more popular neighborhoods have lots of places just selling fruit and fruit juice/smoothies, often selling ice cream too. The fruit in Colombia is outstanding, and the juice bars are unbelievably cheap.
And there's always Crepes & Waffles, a ubiquitous Bogotá chain that—with such a focus—can't help but be great.
There are a few dedicated gourmet zone, the most impressive of which is Zona G (G for Gourmet). It's a quiet, residential-looking neighborhood jam packed with absolutely incredible, world-class restaurants. Other places to look for high-end dining are (naturally) the Zona Rosa, as well as Parque 93, the La Macarena neighborhood of Santa Fé, and a little further afield in Usaquén.
For dining with a view, there are two restaurants up at Monserrate that are not at all tourist traps—they are excellent, modern, high-end restaurants. Just outside the city on the road to La Calera is Tramonti, another mountaintop restaurant less-known to tourists, but done up like a Swiss mountain chalet and perfect for watching the sunset and the lights come on.
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