From the Super Mario Wiki
Boomerangs are a type of weapon that have made numerous appearances throughout the Mario series. When thrown, a boomerang will fly forward, harming any foes who happen to be in its path, then circle back to the user. They are the weapon of choice of the Boomerang Bros., but can also be used by Mario and Luigi after they use the Boomerang Flower.
The first appearance of the boomerang in the Mario series was in The Super Mario Bros. Super Show! episode "Crocodile Mario". Set in the Australia-themed locale of Down Under Land, Koopa Troopas used boomerangs to attack Mario and Luigi as they rafted down a river, but failed to hit them.
Super Mario series
Boomerang Bros. first appear in Super Mario Bros. 3, more specifically in World 1-3, marking the first appearance of boomerangs in a Mario game. Boomerang Bros. mostly appear throughout Desert Land, with two appearing throughout Castle of Koopa. Two Boomerang Bros. also appear at the end of every Treasure Ship, serving as a mini-boss duo.
A special blue Boomerang makes an appearance in some of the e-Reader levels of Super Mario Advance 4: Super Mario Bros. 3. When thrown, the blue Boomerang defeats any enemy it touches before returning to Mario, who much catch it using the B button if he hopes to re-use it. While a few Japan-only levels featured blue Boomerangs, only one of these was made available in other regions: the World-e level Slidin' the Slopes. Near the end is a Boomerang Bro. with a blue Boomerang that is dropped when the enemy is defeated; the player can then pick it up and use it for the remainder of the level, though it disappears upon completion, much like a Goomba's Shoe.
There was also an e-Reader card released only in Japan that uploaded a Boomerang to the player's inventory,
Boomerang Mario is a new form Mario and Luigi can don in Super Mario 3D Land by use of a Boomerang Flower, It gives them the power to throw boomerangs. These boomerangs function much like the blue ones in Super Mario Advance 4, only Boomerang Mario does not need to catch returning boomerangs; he can "produce" an endless amount, although he is limited to using one at a time, and if one is destroyed, he must briefly wait before he can throw another (a glowing from Mario's hands signifies that he can use another). Fire Mario can use a fireball to burn a boomerang that Boomerang Mario has thrown.
In addition to attacking enemies, boomerangs can be used to collect items like coins and Star Medals, as well as hitting blocks. Additionally, if the boomerang hits any five objects excluding blocks and coins consecutively, the player earns a 1-Up.
Pom Pom, a boss introduced in Super Mario 3D Land, also attacks by throwing boomerangs.
During the final two phases of minigame The Final Battle!, Bowser will try to attack the player by throwing fiery boomerangs at them. This is the only known instance of Bowser using boomerangs in the Mario series.
Mario Kart 8
Boomerangs are the object that appears as a launchable item when the Boomerang Flower is used. The player will though hold a Boomerang instead of the flower. They, once throwed, will return back to the player and can be launched up to 3 times.
Boomerangs appear as a new item in Super Smash Bros. for Nintendo 3DS and will also appear in Super Smash Bros. for Wii U. They function like standard boomerangs in that they are thrown to attack opponents and return to the thrower, though for every time the player catches it the boomerang's attack becomes stronger. Other players can catch the boomerang after it is thrown, however.
Boomerangs have been used by Boomerang Bros. in all of their appearances to date. See the Boomerang Bro. article for a more extensive list.