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We study Dillon-type vectorial bent functions of the monomial and multinomial varieties. We also study Kloosterman sums, which relate to the construction of Dillon-type monomial bent functions. For Dillon-type monomial functions we give sufficient conditions for vectorial bentness, leading to the construction of several new examples. We give useful necessary conditions for functions from GF(2^(4m)) to GF(4). We give new restrictions on the maximum output dimension of Dillon-type monomial bent functions on GF(2^(4m)). We subsequently show that certain Dillon-type multinomial bent functions do not meet the Nyberg bound. We give computational results regarding Dillon-type functions from GF(2^(4m)) to GF(4), suggesting that while bent monomials of this type appear to be rare, their binomial counterparts seem to be relatively abundant. Finally, we give divisibility results on Kloosterman sums valued on cosets of certain subfields of GF(2^m), leading to explicit constructions of Kloosterman zeros and Dillon-type monomial bent functions.