Saudade

Saudade (European Portuguese: [sɐwˈðaðɨ], Brazilian Portuguese: [sawˈdadi] or [sawˈdadʒi], Galician: [sawˈðaðe]; plural saudades) is a Portuguese and Galician word that has no direct translation in English. It describes a deep emotional state of nostalgic or deeply melancholic longing for an absent something or someone that one loves. Moreover, it often carries a repressed knowledge that the object of longing will never return. A stronger form of saudade may be felt towards people and things whose whereabouts are unknown, such as a lost lover, or a family member who has gone missing.
Saudade was once described as "the love that remains" after someone is gone. Saudade is the recollection of feelings, experiences, places or events that once brought excitement, pleasure, well-being, which now triggers the senses and makes one live again. It can be described as an emptiness, like someone (e.g., one's children, parents, sibling, grandparents, friends, pets) or something (e.g., places, things one used to do in childhood, or other activities performed in the past) that should be there in a particular moment is missing, and the individual feels this absence. It brings sad and happy feelings all together, sadness for missing and happiness for having experienced the feeling.
In Portuguese, "Tenho saudades tuas" (European Portuguese) or "Tenho saudades de você" (Brazilian Portuguese), translates as "I have saudade of you" meaning "I miss you", but carries a much stronger tone. In fact, one can have saudade of someone whom one is with, but have some feeling of loss towards the past or the future.
In Brazil, the day of saudade is officially celebrated on January 30.

Saudade (1899), by Almeida Júnior