Pokémon Mystery Dungeon

Unlike previous games, evolution in Mystery Dungeon seems nothing more than to get certain Pokémon, in other words, to fill your Friend Areas with every single one of the 386 species. The interesting thing is all you get from evolution in this game is an altered appearance, and for some an additional type or ability to learn different attacks, but evolution does not raise your stats like it drastically does in previous games. To the Pokémon inhabitants in Mystery Dungeon, evolution seems to be a new concept to them as well, as it is unheard of by the characters until Rayquaza has stopped the meteor, so evolution is in other words impossible before you have beaten the first half of the game and seen the credits roll. After the credits have rolled, a part of the shattered meteor has fallen into the Whishcash Pond north of the Pokémon Square and created the Light Cave. It is where all your evolutions take place. However, you can only enter the Light Cave if you are on your own, which means your partner has to go, but when? After you have tried in vain to visit the Light Cave once, and perhaps obtained the Dive HM from Whiscash, when you return to your headquarters, your partner will tell you that it may not be helpful sticking around all the time since you will need the help of other companions, and it will return to its own Friend Area. Now, you may enter the Light Cave and start your evolutions.
Whenever you enter the Light Cave, the glowing stone will ask if you want to evolve, click yes, and it will ask if you want to evolve without items, or with items. Depending how your specie evolves, select accordingly. Note that you must leave the cave and come back again if you want to further evolve, provided that you are at the required level.
You can evolve your other Pokémon too by the 'Leader Switch' option. Go to that particular Pokémon's Friend Area, talk to it, first select the first option to make it join your team. After you do so, talk to it again and pick the second option which is the leader option, this makes that particular Pokémon your leading one.
Remember to send back the rest of your team to their Friend Areas (either go to their Friend Area to do that or by your menu, check the Main Menu section), and you may access the Light Cave for evolution.
The thing about evolution in Mystery Dungeon is that moves are only learnt when you gain a level, not the instant a Pokémon evolves. Take Charizard as an example. Normally it learns Wing Attack at level 36, but you can't get a Charmeleon to evolve until it is at level 36, so when you evolve it, Charizard is already at level 36 and evolution alone will not make you learn the move at that level. There are two ways to make your evolved Charizard relearn Wing Attack in this case. One, there is an item called the Unfortunate Seed, eat it and your level will decrease by one. Say for Charizard again, you return to level 35 after eating one Unfortunate Seed, so when you level up to 36, you will get to learn Wing Attack.
The easier way will be to go to Gulpin's Connect Shop at the Pokémon Square, where move relearning to possible. Select the Pokémon of your choice, then the last option. You will see a list of its old move to learn from. Like when you choose to connect your moves, Gulpin charges for a fixed fee for move relearning.
Some Pokémon must be evolved with special items like in the previous games. In the Light Cave, when it asks you whether you want to use an item, click yes (second option) and you can select the item from your item bag.
After you do so, you will be asked again if you want to proceed with the evolution, or select another item. For Pokémon that evolve with one item, select proceed (first option). Otherwise, attach a second item (second option) and it should automatically proceed to evolution.

The following special items or means are needed for certain evolutions:

Fire Stone - Fire Great Land

Base No.	Pic	Base Name	Final No.	Pic	Final Name
#037		Vulpix	#038		Ninetales
#058		Growlithe	#058		Arcanine
#133		Eevee	#136		Flareon

Leaf Stone - Wish Cave

Base No.	Pic	Base Name	Final No.	Pic	Final Name
#044		Gloom	#045		Vileplume
#070		Weepinbell	#071		Victreebel
#102		Exeggcute	#103		Exeggutor
#274		Nuzleaf	#275		Shiftry

Moon Stone - Sun Cave

Base No.	Pic	Base Name	Final No.	Pic	Final Name
#030		Nidorina	#031		Nidoqueen
#033		Nidorino	#034		Nidoking
#035		Clefairy	#036		Clefable
#039		Jigglypuff	#040		Wigglytuff
#300		Skitty	#30		Delcatty

Sun Stone - Sun Cave

Base No.	Pic	Base Name	Final No.	Pic	Final Name
#044		Gloom	#182		Bellossom
#191		Sunkern	#192		Sunflora

Thunder Stone - Lightning Great Land

Base No.	Pic	Base Name	Final No.	Pic	Final Name
#025		Pikachu	#026		Raichu
#133		Eevee	#135		Jolteon

Water Stone - Northwind Great Land

Base No.	Pic	Base Name	Final No.	Pic	Final Name
#061		Poliwhirl	#062		Poliwrath
#090		Shellder	#091		Cloyster
#120		Staryu	#121		Starmie
#133		Eevee	#134		Vaporeon
#271		Lombre	#272		Ludicolo

Link Cable - in place of HMs or other items when you already have the item (Sun Cave, Flying Dragon Hole)

Base No.	Pic	Base Name	Final No.	Pic	Final Name
#064		Kadabra	#065		Alakazam
#067		Machoke	#068		Machamp
#075		Graveler	#076		Golem
#093		Haunter	#094		Gengar

Link Cable + Deep Sea Scale - Big Sea

Base No.	Pic	Base Name	Final No.	Pic	Final Name
#366		Clamperl	#368		Gorebyss

Link Cable + Deep Sea Tooth - Big Sea

Base No.	Pic	Base Name	Final No.	Pic	Final Name
#366		Clamperl	#368		Huntail

Link Cable + King's Rock - Wish Cave

Base No.	Pic	Base Name	Final No.	Pic	Final Name
#061		Poliwhirl	#186		Politoed
#079		Slowpoke	#199		Slowking

Link Cable + Dragon Scale - Flying Dragon Hole

Base No.	Pic	Base Name	Final No.	Pic	Final Name
#117		Seadra	#230		Kingdra

Link Cable + Metal Coat - Southern Hole

Base No.	Pic	Base Name	Final No.	Pic	Final Name
#095		Onix	#208		Steelix
#123		Scyther	#212		Scizor

Link Cable + Upgrade - Wish Cave

Base No.	Pic	Base Name	Final No.	Pic	Final Name
#137		Porygon	#233		Porygon2

Smartness (four stars or above)

Base No.	Pic	Base Name	Final No.	Pic	Final Name
#042		Golbat	#169		Crobat
#113		Chansey	#242		Blissey
#172		Pichu	#025		Pikachu
#173		Cleffa	#035		Clefairy
#174		Igglybuff	#039		Jigglypuff
#175		Togepi	#176		Togetic
#298		Azurill	#183		Marill