My new toy
Build the first car.


Oops, it happened by accident
Defeated your first opponent.


Misanthrope
Defeat 10 opponents.


Eliminator
Defeat 33 opponents.


I'm just getting started
Finish in 1st place 25 times.


Fetishist
Finish in 1st place 50 times.


Fallen KGB
Defeated Ivan Komrad.


Extra dip
Defeated Big Cheese.


Last nerf
Defeated Yuji Than.


Drunk driver
Defeated Joe Boomer.


I call the police
Defeated Sonya Mass.


Rage and passion
Defeated Frau Uber.


Enemy of life
Defeated Frank Woods.


Self-explanatory. You should obtain them while completing the career mode and other achievements.

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Valkyrie
Fly out of your windshield 10 times.


Catapult
Fly out of your windshield 33 times.


You can grind it in Bowling, Darts or Minigolf games.

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The punisher of property
Destroy 100,000 world objects.


The requirement has been increased from 1,000 to 100,000. You can grind it on tracks with many props, such as Old Village or Village Outskirts.

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Toss a coin to your Mihalych
Take money from Zhenka when you don't have much money.


Spend all your money on cars and upgrades. When you do it, you'll see a notification that Zhenka wants to give you some extra money.

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Auto mechanic
Purchase and build 7 vehicles.


Collector
Purchase and build 14 vehicles.


Dump king
Purchase and build 21 vehicles.


To spend as little money as possible, buy only the cheapest cars. You need to have all 21 cars at once. You can also help yourself with Moto Rush DLC, which adds several very cheap vehicles.

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Interstellar
Reach a top speed of 300 km/h / 186 mph.


Select Cross -> Autódromo, then select a fast car from the A class and use all your nitro on the straight.

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Get down!
Knock a standing opponent to the ground.


Sometimes when you destroy the opponent's car, they'll get out and stand on the road. Just run over them and you'll get this achievement. I recommend to do it on shorter tracks, e.g. Cross -> Cycle.

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I'm the one with the f*cking ***!
Eliminate all opponents in a Rush or Cross event, and the finish in first place.


Select a slow and weak car, but don't upgrade it. Then, Select Mono.Class -> Cycle and set 15 laps (or more if you wish). Drive the opposite way, ram the first opponent from the front and try to block them so the other opponents would ram them as well. When you have low health, change the tactics - drive after the opponents and ram them from the back or the side.

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Perfectionist
Completed the story career 100%.


You need to win all races in the career mode.

Dead, but first
Take the first place with the driver's body.


Select Rush -> S.F. Hills and set one lap. You need to crash at the arch, so the driver's body will cross the finish line. Although the description states that you have to be first, actually you don't have to. I came 5th when I got this achievement.

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The leap of fate
Make the legendary jump in the canals in the specific car.


Select Rush -> L.A. Streets and Bigsmoker T1 (to unlock it, you have to complete 'Dump King' achievement and claim the reward in FestApp). You need to jump from the bridge shown on the screenshots and recreate the scene from Terminator 2 movie.

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The real geek
Visit the five famous places on the big maps.


#1 Select Rush -> Behind The Port

#2 Select Rush -> L.A. Rundtown

#3 Select Rush -> Cargo Port

#4 Select Rush -> Grand Canyon

#5 Select Rush -> Grand Canyon

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Hacker bummer
Resist a hacker attack in Dash mode.


Pick up the Impulse powerup (pink circle icon) and use it when an opponent is about to hack you. The opponent must be very close to you. Some people say that they did it by blocking the hack attack with a shield, but for me that didn't work.

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Officer Mihalych
Eliminate 300 candidates in Hunter mode.


Choose Triumph (the shortest track available) and Responder (the most durable police car). Keep spamming mines and spike strips through the entire race - they will never despawn by themselves and the candidates will eventually hit each of them. First, get 6 kills to summon a helicopter, which will shoot at candidates by itself. Then, get another 9 kills to get a drone, which you can control by pressing H (WASD + mouse - flying, LMB - machine gun, RMB - rockets). When you are low on health, get 12 kills to get a new police car (you can do it only once per race).

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**** ***!
Get a ticket for parking in downtown San Francisco under a no parking sign.


Select Rush -> S.F. Downtown. Next to the mall turn left and stop behind the barriers, where no parking sign is located. After several seconds you'll receive $1,000 ticket.

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I'll get you even in hell
Kill Frank Woods in Dead Out mode.


It's very easy - Frank is just next to the hideout. Run him over to get this achievement.

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Fashionista
Find 30 outfits in the game.


You just need to unlock 30 outfits in your wardrobe, no need to buy them. You can unlock them by:

- completing achievements and boss races;

- using cheat codes: PROLOGUE, FROZEN, STIG, KNIGHT (type them on your license plate, they don't affect achievements);

- buying Moto Rush DLC (adds 5 skins);

- finding them on tracks from Wild Roads update (The Beach Way, Madagascar's Heart, Apocalypsis Now);
https://steamproxy.net/sharedfiles/filedetails/?id=2990307872
- finding them on tracks from boss races.
https://steamproxy.net/sharedfiles/filedetails/?id=2863473752
After 5.0 update the race tracks for Yuji Than and Frau Uber have been changed and the guide I linked above hasn't been updated. Here are new locations of the outfits.

Yuji Than race

Frau Uber race

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I am Mihalich
Finish the game in NEW GAME + mode.


After completing the campaign mode you'll unlock New Game + mode and basically you'll have to complete the campaign once again. Previously this achievement was about obtaining all other achievements.

The usual thing
Run over a spectator.


You can often see spectators standing on the road. If you run over them, you won't earn any new fans for a short period of time.

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Driving dead
Finish a Race or Derby event with 1HP remaining.


Select any race and keep damaging your car until the yellow health bar is almost empty. Then, drive carefully behind the opponents and finish the race.

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Pacifist
Complete a race or cross without a single damage to the car.


Select Cross -> Autodrómo, set 1 lap and drive behind the opponents till the end of the race.

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Casino always wins
Win $50,000 in Roulette mode.


After beating Ivan Komrad you'll get access to the Roulette mode. After some testing I figured out that the most profitable way is to go to Pick me option and select J&H Racing or Boss Helicopter mode. J&H Racing only adds two more opponents (Joker and Harley), so it's an easier option, but you'll have to win 13 races to get $50k (entry fee is $2k and reward for the 1st place is $4k). Boss Helicopter flies in front of you and drops random traps, which may be tricky to avoid on some tracks, so it's a harder option, but you'll have to win only 7 races (entry fee is $4k and reward for the 1st place is $8k).
I don't recommend choosing other modes. Drunk driver makes the camera float around and with Car shuffle the game may keep giving you slow cars, so winning the race is much harder. Mix-up adds all the features together, which makes the race impossible to win.
You can lose any amount of money you want. The only thing that matters here is money won from races. You can see how much you won on the bottom of the Roulette menu.

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Kind driver
Make a few donations.


Go to FestApp -> Donations. Instead of claiming donations to yourself, click Give to charity button. I'm not sure what counts here: amount of money donated or a number of donations. I made about 10 donations when I got this achievement.

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Too close
Hand off the bomb in contact mode at the last second.


Pick up a bomb, then follow one of the opponents and when the timer shows 1 second, pass it to them. If they somehow manage pass the bomb back to you before the explosion, that will still be valid for the achievement. I recommend to do it when there are 4-5 opponents - there won't be too much chaos on the arena and you will still be able to target one of them easily.

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Forgiveness
Meet up with Zhenka despite the fight in the plot company.


Beat Frau Uber, then you'll see a cutscene, where Zhenka leaves Mihalych due to a quarrel. After that, wait for a donation of $5 from Zhenka himself.
Go to Free Race -> Rush -> Tokyo Districts. Then, find a pink tree, where Zhenka's car is parked. You'll see it after you drive off the highway. When you drive to him, a cutscene will play.

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Special thanks to:
Köströv, ilya515 and S.r-Billy for making guides in Russian and Portuguese
Hantik_Lonberg for Forgiveness achievement guide
SonicUnleashedXY and DΞX STΛRR for outfits guide
MESYA for cheat codes guide