  Weighted projective space arises when we consider the usual geometric
definition for projective space and allow for non-trivial weights. On its own,
this extra freedom gives rise to more than enough interesting phenomena, but it
is the fact that weighted projective space arises naturally in the context of
classical algebraic geometry that can be surprising. Using the Riemann-Roch
theorem to calculate l(E,nD) where E is a non-singular cubic curve inside
projective 2-space and D=p is a point in E we obtain a non-negatively graded
ring R(E) by taking the direct sum of the L(E,nD) for n greater than or equal
to 0. This gives rise to an embedding of E inside the weighted projective space
P(1,2,3).
  To understand a space it is always a good idea to look at the things inside
it. The main content of this paper is the introduction and explanation of many
basic concepts of weighted projective space and its varieties. There are
already many brilliant texts on the topic but none of them are aimed at an
audience with only an undergraduate's knowledge of mathematics. This paper
hopes to partially fill this gap whilst maintaining a good balance between
'interesting' and 'simple'.
  The main result of this paper is a reasonably simple degree-genus formula for
non-singular 'sufficiently general' plane curves, proved using not much more
than the Riemann-Hurwitz formula.
