How to Find Giovanni

Finding Giovanni isn't like finding the other Team GO Rocket Leaders (Arlo, Cliff, and Sierra) who are discovered by collecting mysterious components from regular Team GO Rocket Grunts. Those mysterious components get combined into a Rocket Radar. In order to find Giovanni, you'll need a Super Rocket Radar. Here's how to get it.

https://www.thegamer.com/pokemon-go-giovanni-location/

Prerequisite: Complete The "A Troubling Situation" Special Research Quest

The special research quest "A Troubling Situation" needs to be completed before you can find Giovanni. You only receive this research quest once.

It has four steps that are designed to introduce you to the Team Go Rocket encounters.

Step One Tasks

• Evolve Two Pokemon (Reward: Murkrow Encounter)
• Power-Up A Pokemon Five Times (Reward: Sableye Encounter)
• Reward For Completing All Step One Tasks: 1,000 Stardust and 2,000 XP

Step Two Tasks

• Catch Two Shadow Pokemon (Reward: Koffing Encounter)
• Defeat Three Team GO Rocket Grunts (Reward: Ekans Encounter)
• Reward For Completing All Step Two Tasks: 2,000 Stardust, 2,000 XP, Five Golden Razz Berries

Step Three Tasks

• Purify Three Shadow Pokemon (Reward: 3,000 Stardust)
• Purify a Fire, Water, or Grass-Type Shadow Pokemon (Reward: 3,000 XP)
• Reward For Completing All Step Three Tasks: 2,000 Stardust, One Rare Candy, Five Silver Pinap Berries

Step Four Tasks

• Auto-Completed Task (Reward: 1,000 Stardust)
• Auto-Completed Task (Reward: 1,000 XP)
• Auto-Completed Task (Reward: 1,000 Stardust)
• Reward For Completing All Step Four Tasks: One Fast TM, One Charged TM, Three Rare Candies.

if you pop a Lucky Egg and Star Piece before completing steps three and four, you'll get double the amount of XP and  50% more of stardust. 

How The Monthly Special Research To Find Giovanni Works

Looming in the Shadows

Once you've completed the "A Troubling Situation" special research quest, you're eligible to receive the monthly quest to hunt down Giovanni.

Each month you can receive a special research quest (which has a different name each time) that leads to beating Giovanni. The first few steps differ from month to month, but the fourth, fifth, and sixth steps are always the same.

Giovanni can only be fought once per month. If you complete one month's research in the following month (i.e. you complete the January research in February), you'll receive the February research immediately after completing the January research. If you complete the February research in February, you'll need to wait until March to get the research again.

Step Four: Fight The Team GO Rocket Leaders

• Defeat Team GO Rocket Leader Arlo (Reward: 1,250 XP)
• Defeat Team GO Rocket Leader Cliff (Reward: 1,250 XP)
• Defeat Team GO Rocket Leader Sierra (Reward: 1,250 XP)

Reward For Completing All Step Four Tasks: Three Golden Razz Berries, 2,000 Stardust, One Super Rocket Radar.

To find the Team GO Rocket Leaders, you'll need to fight Team GO Rocket Grunts. Beating each Grunt in battle will provide you with one mysterious component and six mysterious components are needed to create a Rocket Radar.

When you have a Rocket Radar equipped, Shadow PokeStops can be occupied by a Leader instead of a Grunt. You'll hold on to the Rocket Radar until you beat a Leader in battle.

Once you do, the Rocket Radar disappears and you'll need to collect six more mysterious components to challenge another Leader. After you've beaten all three leaders, you'll earn the Super Rocket Radar - it works just like the regular kind, except it will help you find Giovanni instead of the three Leaders.

Step Five: Challenge Giovanni And Win

Now that you have the Super Rocket Radar, you'll be able to search for Giovanni. The Shadow PokeStops will look like they have Giovanni at them from the map but when you click on them you'll discover that it was actually a Grunt in disguise. You'll need to keep checking PokeStops until you uncover the real Giovanni.

• Find the Team GO Rocket Boss (Reward: 2,500 Stardust)
• Battle the Team GO Rocket Boss (Reward: 1,500 XP)
• Defeat the Team GO Rocket Boss (Reward: Three Silver Pinap Berries)

Once you defeat Giovanni in battle (which is quite tough, so go in prepared!), you'll have a chance to capture his special Shadow Pokemon.

Reward For Completing All Step Five Tasks: One Charged TM, One Fast TM, 3,000 Stardust.

Step Six: Reap The Rewards Of Your Success

• Auto-Completed Task (Reward: 2,000 XP)
• Auto-Completed Task (Reward: 2,000 XP)
• Auto-Completed Task (Reward: 2,000 XP)
• Reward For Completing All Step Six Tasks: Three Rare Candy, Three Max Revives, 20 Ultra Balls.

Congratulations! You've defeated Giovanni and stolen his Pokemon, hopefully in order to purify them and keep them happy in your party. You'll be able to do it all over again once you receive the tasks again.

Can't Find Giovanni?

Don't panic if you can't find Giovanni or don't receive his special research; Pokemon GO has been known to take breaks from it. At the time of this writing, September 2021, Giovanni has disappeared from the game with no indication of when he'll return. Niantic's only comment on the matter has been: “We hope he’s not planning something in the shadows!”

Note: I get an answer from my post in Facebook. There is no monthly Giovanni now. Level 20 can beat Grunt leaders, Arlo, Cliff, Sierra. It matches to your level.