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Cosmopolitan Mexico City can seem somewhat pricey compared to the rest of the nation, but when you factor in all the free activities on offer, it's very possible have a great time on a tight budget.

From first-rate museums and inspiring public art to invigorating hikes and live music, many of the cultural capital's premier attractions won't set you back a single peso. You won't pay much to get around, either – at a mere five pesos a ride, the Metro is a steal.

These are the best ways to explore Mexico City without reaching for your wallet.

1. Admire the murals at Palacio de Bellas Artes

Dazzling frescos painted by the three leading figures of the Mexican mural movement cover the stairwells of the art deco Palace of Fine Arts, including Diego Rivera’s compelling Man, Controller of the Universe and José Clemente Orozco’s vibrant Catharsis. Take advantage of free admission on Sundays.

If the palace's theater happens to be open, duck in for a peek at the glorious stained-glass curtain that was assembled with nearly a million pieces of glass by New York's Tiffany & Co.

2. Take a stroll through Bosque de Chapultepec

Chapultepec Park's 1700 acres of wooded grounds are made for peso-pinching. Here you can gaze at Diego Rivera's larger-than-life mosaic sculpture of the rain god Tlalóc, stroll through a botanical garden with rare orchids and marvel at the Voladores de Papantla (Papantla Flyers) as they enact a high-flying rainmaking ritual in front of the world-class . Or simply pack a torta (sandwich) and plop down in a grassy picnic area below the imposing Chapultepec Castle.

3. Bring your passport for the Palacio Nacional

Not only does Mexico's presidential residence occupy one of the city's most storied buildings (it was built atop a royal Aztec palace), it also houses more Diego Rivera murals – dramatic renderings that visually narrate the nation's most transformative class struggles, from the brutal Spanish conquest up to the 20th-century post-revolutionary period, when the artist's masterpiece was completed. You'll need your passport to access the building.

The National Palace is also home to the Museo Nacional de las Culturas, a free museum that exhibits the art, dress and handicrafts of the world's cultures.

Planning tip: The Palacio Nacional is a working government building and will sometimes close to the public if there are private functions – check before your visit to be sure it's open when you want to visit.

4. Explore Mexico City on two wheels

Riding a bike in the chaotic capital can feel like an adventure sport at times but on Sundays, the broad, tree-lined Avenida Paseo de la Reforma is closed to traffic from 8am to 2pm, making it a relatively carefree ride between Chapultepec Park and the city's Historic Center.

“Bicigratis” kiosks near the Angel of Independence, Chapultepec Park's Anthropology Museum, and several booths dotting the Roma, Condesa and Centro Histórico neighborhoods loan free bikes that you can take out for three hours if you leave a passport or driver's license with the operator.

Planning tip: The free bikes are on a first-come-first-served basis, so you may miss out. on Paseo de la Reforma offers hourly rental and bilingual guides.

5. Hike the city's highest peak at Parque Nacional Cumbres del Ajusco

Mexico City may not conjure up images of a hiking destination, but you should never underestimate the capital's capacity to surprise.

For an exhilarating climb up the city's highest peak, head for the fragrant pine forest of Ajusco National Park and hike to Pico del Aguila, the summit of an extinct volcano range that affords expansive views of the Valley of Mexico. Try to visit on a weekday, when the park has far fewer guests.

6. Browse the artistic offerings at Museo Jumex and Museo Soumaya

The exhibits and architectural styles of these adjacent museums in the affluent Polanco district couldn't be more different. On one end of Plaza Carso, you have the Museo Jumex, which showcases cutting-edge contemporary art in a minimalist building designed by English architect David Chipperfield.

Meanwhile, just across the plaza, you can't miss the shiny Museo Soumaya, a six-story building plated with 16,000 hexagonal aluminum tiles. The free museum holds Mexican billionaire Carlos Slim's massive art collection, a hit-or-miss affair featuring lesser-known works by renowned artists, such as Picasso, Van Gogh and Dalí, and a considerable number of Rodin sculptures.

Planning tip: Both museums are free to enter, but Museo Jumex also offers guided tours at 10am, 1pm and 2pm, Tuesday to Friday for M$500 Pesos (US$24.80) per person. They are very popular – arrive early if you want a spot.

7. Experience the underground scene at Tianguis Cultural del Chopo

And you thought punk was dead. Well, it's very much alive and kicking at this Saturday flea market, fondly known as El Chopo. Every week, mohawked punks, tatted headbangers and avid stoners gather at this tianguis (outdoor market) to browse for cult videos, rare music and vintage clothing as young-and-hungry bands jam on the market's rear stage. It's a cool spot to tap into the city's underground scene and get word of live music events.

Planning tip: Keep an eye on your bag – pickpockets are known to target music fans who are focused on the band.

8. Take a self-guided tour of the historic city center

With more than 1500 historic buildings, some as old as the city itself, a stroll through downtown's Centro Histórico – a UNESCO World Heritage Site – offers a crash course in Mexico's past.

Start by contemplating the ancient ruins of Templo Mayor, a sacred temple that once stood at the heart of the Aztec Empire. Aztec dances are still practiced there today. Next, take in the colonial architecture on the nearby ��������. The cathedral and presidential palace were constructed some 500 years ago by Spanish colonizers using bricks they'd stolen from destroyed Aztec buildings.

After that, wander a half dozen blocks west to Alameda Central, the oldest urban park in the Americas and home to a famous Diego Rivera mural in Museo Mural Diego Rivera, which you can appreciate for free on Sunday. On Alameda's east side, send off a postcard in the Palacio Postal, a golden palace for the ages.

9. Make a pilgrimage to Basílica de Guadalupe

Legend has it that in 1531, the Virgin of Guadalupe – aka the Virgin Mary – appeared four times to a peasant named Juan Diego up on a sacred hill near the present-day Basílica de Guadalupe.

According to Diego's account, after the last sighting, her image was miraculously emblazoned on his cloak, which now hangs behind the church's main altar and can be seen up close along a surreal moving walkway. The basilica is one of the world's most important Catholic pilgrimage sites, especially on December 12, when it sees up to a million visitors for the feast day of Our Lady of Guadalupe.

10. Listen to free music in Plaza Garibaldi

It doesn't cost a thing to stand in the background and listen to the hired musicians working the crowds at Plaza Garibaldi, Mexico City's famed mariachi square.

The mariachi ensembles take center stage, but you'll also hear regional musicians belting out the classics, such as white-clad son jarocho trios hailing from Veracruz and accordion-driven �Դǰ��ٱ�ñ�� bands from northern Mexico.

Planning tip: The atmospheric plaza really comes to life at night, and although the square itself is considered relatively safe, avoid exploring Garibaldi's dodgy periphery after dark.

11. Experience the sinking city phenomenon

Mexico City is sinking fast, due mostly to excessive groundwater extraction. Experts estimate that some areas of the city sink as much as 51cm (20 inches) a year as the water table continues to drop.

To fully grasp the capital's sinking-city phenomenon, check out the slanted Catedral Metropolitana, Mexico City’s iconic cathedral on the main square, then walk one block east to the free Ex Teresa Arte Actual museum, a former convent with incredibly inclined floors. Even the Angel of Independence monument on Avenida Reforma, yet another important landmark, requires regular repairs due to ongoing sinking and earthquakes.