  Let $X$ be a normal projective variety over an algebraically closed field of
characteristic zero. Let $D$ be a reduced Weil divisor on $X$. Let $G$ be a
reductive linear algebraic group. We introduce the notion of a logarithmic
connection on a principal $G$-bundle over $X$, which is singular along $D$. The
existence of a logarithmic connection on the frame bundle associated with a
vector bundle over $X$ is shown to be equivalent to the existence of a
logarithmic covariant derivative on the vector bundle if the logarithmic
tangent sheaf of $X$ is locally free. Additionally, when the algebraic group
$G$ is semisimple, we show that a principal $G$-bundle admits a logarithmic
connection if and only if the associated adjoint bundle admits one. We also
prove that the existence of a logarithmic connection on a principal bundle over
a toric variety, singular along the boundary divisor, is equivalent to the
existence of a torus equivariant structure on the bundle.
