  One often distinguishes between a line and a plane by saying that the former
is one-dimensional while the latter is two. But, what does it mean for an
object to have $d-$dimensions? Can we define a consistent notion of dimension
rigorously for arbitrary objects, say a snowflake, perhaps? And must the
dimension always be integer-valued? After highlighting some crucial problems
that one encounters while defining a sensible notion of dimension for a certain
class of objects, we attempt to answer the above questions by exploring the
concept of Hausdorff dimension -- a remarkable method of assigning dimension to
subsets of arbitrary metric spaces. In order to properly formulate the
definition and properties of the Hausdorff dimension, we review the critical
measure-theoretic terminology beforehand. Finally, we discuss the notion of
self-similarity and show how it often defies our quotidian intuition that
dimension must always be integer-valued.
