HT Games
You're almost certainly thinking of **Pokémon**—the franchise built on catching, training, and battling monsters.
| Game | Year | Generation | Key Feature |
|---|---|---|---|
| Diamond / Pearl | 2006/2007 | Gen 4 (Sinnoh) | First DS entry, Wi-Fi trading/battling |
| Platinum | 2008/2009 | Gen 4 | Enhanced story, Battle Frontier, Giratina forms |
| HeartGold / SoulSilver | 2009/2010 | Gen 2 Remake | Johto + Kanto, Pokéwalker support |
| Black / White | 2010/2011 | Gen 5 (Unova) | Fully animated sprites, seasons, triple battles |
| Black 2 / White 2 | 2012 | Gen 5 Sequels | New story, Key System, Pokéstar Studios |
If it had "Gotta Catch 'Em All" vibe, it's Pokémon.