  In this thesis, we explore the subject of complex spacetimes, in which the
mathematical theory of complex manifolds gets modified for application to
General Relativity. We will also explore the mysterious Newman-Janis trick,
which is an elementary and quite short method to obtain the Kerr black hole
from the Schwarzschild black hole through the use of complex variables. This
exposition will cover variations of the Newman-Janis trick, partial
explanations, as well as original contributions
