General happiness:
* Joy
* Contentment
* Bliss
* Euphoria
* Cheerfulness
* Delight
* Gladness
* Exuberance
* Fulfillment
More specific happiness:
* For a specific reason: Thrilled, elated, overjoyed, ecstatic
* Inner peace: Tranquility, serenity, peace of mind
* Feeling good physically: Well-being, comfort, ease
* Being satisfied: Contentment, gratification
Figurative or poetic synonyms:
* Rapture
* Ecstasy
* Beatitude
* Elysium
The best synonym depends on the context and the specific type of happiness you want to express.