Abelian varieties are algebraizable complex tori. As such they carry polarizations, i.e., special codimension 1 subvarieties) that can be used to embed them in projective space. Theta divisors are minimal among such divisors, in the sense that there is only one holomorphic function (up to a scalar) with poles of order 1 solely along the theta divisor. This property has some interesting consequences for the cohomology of the theta divisor: the primal (or vanishing) cohomology of the theta divisor behaves like a Hodge structure of lower weight. I will survey some results about interesting geometric consequences of this fact for abelian varieties of low dimensions.