  A Besicovitch set is a subset of $\R^d$ that contains a unit line segment in
every direction and the famous Kakeya conjecture states that Besicovitch sets
should have full dimension. We provide a number of results in support of this
conjecture in a variety of contexts. Our proofs are simple and aim to give an
intuitive feel for the problem. For example, we give a very simple proof that
the packing and lower box-counting dimension of any Besicovitch set is at least
$(d+1)/2$ (better estimates are available in the literature). We also study the
`generic validity' of the Kakeya conjecture in the setting of Baire Category
and prove that typical Besicovitch sets have full upper box-counting dimension.
  We also study a weaker version of the Kakeya problem where unit line segments
are replaced by half-infinite lines. We prove that such `half-extended
Besicovitch sets' have full Assouad dimension. This can be viewed as full
resolution of a (much weakened) version of the Kakeya problem.
