Just Cause 2

Challenges are races. They will pop up on your map inside faction territory. This means that in order to have access to the challenges you first need to do takeover missions for the factions. Once done, navigate to your PDA map. You will notice a flag symbol. Those are the challenges who amount to a total of 75.

Approach the location and look for a flag post that is at eye level. You do not need a vehicle to start the race as it will be provided. There are different types of races requiring different vehicles and different skill levels. You will see all this information on a confirmation screen once you pushed near the flag post.

The actual races are checkpoint races. For every gate you will get a time bonus. Your finishing time does not matter as long as you don't run out of time. Vehicles can be wrecked during the race as long as you can replace them before you have not enough time left to finish the challenge. Take note of the vehicle type. You will get a message at the start of the race informing you explicitly that only specific vehicles can be used. Only those can be used to cross the checkpoints. The categories are air, road, water and foot.

You can find the provided vehicles near the flag post. They will be easy to spot most of the time. Take your time and look around as most of the time they are best for completing the challenge.

The race will start once you cross the starting line. Check out Solution for Up to the Challenge 2 In Just Cause 2 for more tips on difficult challenges. Good luck!

Review 24 September 2014: Added button icon.

In addition to the very good tips already submitted:

Road challenges:
I found using a motorcycle (Hamaya Cougar 600 is superb and spawns regularly) for road challenges is usually is a better option than the sports car that spawns. I would use the default vehicle if it is an off-road challenge (beach or dirt track)

Base jump challenges:
It took me a while to figure out that you'll need to press forward on LS as you jump to hit the first few rings on these. If you miss a ring and need to retry, hit start and select Restart from Checkpoint to get you back on top of the (usually very tall) building quickly.

Air challenges:
Use a helicopter when you can, especially around the Panau City areas. If you need something faster, the prop plane that spawns is usually the best bet. If a Cassius private jet spawns instead, you may prefer to buy a Pell Silberbolt from the black market instead (preferably 4-star or above)

I got this in Mercenary mode and opted for road races. These tend to be much easier as both the chopper and plane challenges are over if you crash and the foot races are over if you miss a checkpoint (it's hard to get back to where you were if you miss). The boat races are somewhat easier but it's difficult to get another boat that's suitable if you end up on land or your boat explodes. If you're half way through a road race, there's usually cars nearby to hijack if you lose the car.

You can tell what type of challenge it is from the icon. My advice, go for road races and it shouldn't take more than an hour to get the 10 required.