We discuss several enumerative results for irreducible polynomials of a givendegree and pairs of relatively prime polynomials of given degrees in severalvariables over finite fields. Two notions of degree, the {\em total degree} andthe {\em vector degree}, are considered. We show that the number ofirreducibles can be computed recursively by degree and that the number ofrelatively prime pairs can be expressed in terms of the number of irreducibles.We also obtain asymptotic formulas for the number of irreducibles and thenumber of relatively prime pairs. The asymptotic formulas for the number ofirreducibles generalize and improve several previous results by Carlitz, Cohenand Bodin.
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