Hilbert's 17th problem asked whether a polynomial on real Euclidean space, with nonnegative values, must be a sum of squares of rational functions. It was solved affirmatively by E. Artin around 1926. My own work on proper holomorphic mappings in several complex variables led me to seek complex variables analogues. For example, must a nonnegative polynomial on complex Euclidean space be the quotient of squared norms of holomorphic polynomial mappings? The answer in general is no. The answer is yes in certain circumstances, and this leads to a reinterpretation of the question in the language of holomorphic line bundles on complex projective space. This then leads to a general isometric imbedding theorem for holomorphic vector bundles (proved by Catlin and the speaker) related to Calabi's famous result. The first three fourths of the talk will be understandable by beginning graduate students.