The surname Rasmussen is a patronymic name derived from the German personal name Erasmus (which in turn was derived from the Latinized form of the Greek word "erasmos," meaning love), borne by an early Christian saint.  Generally, many of the family evolved the name to Rasmussen in Denmark. The suffix '-sen' is translatable as 'son of.'
Rasmussen Spelling Variations
Spelling variations of this family name include: Rasmussen, Rasmusen, Asmussen, Rasmus, Asmus, Erasmus and many more.
Early Notables of the Rasmussen family (pre 1700)
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Emigration to New Zealand followed in the footsteps of the European explorers, such as Captain Cook (1769-70): first came sealers, whalers, missionaries, and traders. By 1838, the British New Zealand Company had begun buying land from the Maori tribes, and selling it to settlers, and, after the Treaty of Waitangi in 1840, many British families set out on the arduous six month journey from Britain to Aotearoa to start a new life. Early immigrants include:
Rasmussen Settlers in New Zealand in the 19th Century
Maren Rasmussen, aged 22, who arrived in Wellington, New Zealand aboard the ship "Halcione" in 1870
Mrs. Lena Jakobsen Rasmussen (d. 1912), (née Solvang), aged 62, Norwegian Third Class passenger from Haugesund, Rogaland who sailed aboard the RMS Titanic and died in the sinking 
Mr. Arthur Severin Rasmussen, American Carpenter's Mate First Class from California, USA working aboard the ship "USS Arizona" when she sunk during the Japanese attack on Pearl Harbor on 7th December 1941, he died in the sinking 
^ Smith, Eldson Coles, New Dictionary of American Family Names New York, Harper & Row, 1956. Print
^ Filby, P. William, Meyer, Mary K., Passenger and immigration lists index : a guide to published arrival records of about 500,000 passengers who came to the United States and Canada in the seventeenth, eighteenth, and nineteenth centuries. 1982-1985 Cumulated Supplements in Four Volumes Detroit, Mich. : Gale Research Co., 1985, Print (ISBN 0-8103-1795-8)