Purchase "Dino: The Essential Dean Martin" if you wish to own the most complete compilation of his hits in a single source. This collection was released in 2004 by Capitol Records, then re-released in 2005 as a Special Platinum Edition. The 2004 release has 30 digitally remastered songs. Among these are the obvious hits, like "That's Amore" and "Volare," and forgotten hits or ones that Dean himself loved, like "Send Me the Pillow You Dream On."
The Platinum Edition includes a second CD of a live performance Martin recorded at Lake Tahoe in 1962, with an introduction by Sinatra. This is a rare gem. At the time, Martin was under contract with Sinatra's Reprise Records, and the Rat Pack was at its height of popularity.
Buy a few pivotal albums if you wish to compile a more complete discography. Martin was signed with Capitol Records from 1953 to 1964, overlapping his time with Reprise, from 1962 to 1978. (Capitol continues to release and re-release Martin's recordings.) His first album, "Dean Martin Sings," was released at the height of his partnership with Jerry Lewis, largely as a novelty. It contains none of Martin's recognizable hits but is essential for collectors.
"This Time I'm Swingin'" (Capitol, 1960) is considered one of his best albums. It includes the hits "You're Nobody Till Somebody Loves You" and "Just in Time," and a very romantic but rare recording of "I've Grown Accustomed to Her Face." "Hey Brother, Pour the Wine" (Capitol, 1964) has the ebullient "Memories Are Made of This," the romantic "Sway" and "Standing on the Corner," plus some fun if not especially well-known songs.
Purchase some albums from the Rat Pack era, when Martin had the most artistic control over his records and his voice was at its most mature.
Martin was proud of his Italian heritage and recorded "Dino: Italian Love Songs" for Capitol Records in 1962. This includes such significant songs as "Ritorna Me," "Arrivederci, Roma" and "Non Dimenticar." It also includes a recording of "I Have but One Heart," made famous a decade later as the wedding song in "The Godfather."
One gem is "Sleep Warm" (Capitol, 1959), which was orchestrated by Sinatra. This concept album had a bedtime theme, with such recordings as "Cuddle a Little Closer" and "Good Night Sweetheart." "The Rat Pack Live at the Sands" captures a 1963 performance, with Martin acting as host to performances by Sinatra and Sammy Davis Jr. It was not released until 2001 by Capitol.
If you like country music, pick up some of Martin's country albums. His country music recording career started with "Country Style" (Reprise, 1963). He was named the Country Music Association's Man of the Year in 1965, both for his recordings and for featuring country musicians on his TV variety show.
His three best-selling country albums were "Dean 'Tex' Martin" (Reprise, 1963), "Welcome to My World" (Reprise, 1967) and "Gentle on My Mind" (Reprise, 1968). After Warner Bros. Records purchased Reprise, it released an excellent Dean Martin country compilation in 1983, "The Nashville Sessions."