Excursions from a fixed point $b$ are studied in the
framework of a general Borel right process $X$, with a fixed excessive
measure $m$ serving as background measure; such a measure always exists
if $b$ is accessible from every point of the state space of $X$. In this
context the left-continuous moderate Markov dual process
$\widehat X$ arises naturally and plays an important role. This allows
the basic quantities of excursion theory such as the Laplace-L\'evy
exponent of the inverse local time at $b$ and the Laplace transform of
the entrance law for the excursion process to be expressed as inner
products involving simple hitting probabilities and expectations. In
particular if $X$ and
$\widehat X$ are honest, then the resolvent of $X$ may be expressed
entirely in terms of quantities that depend only on $X$ and
$\widehat X$ killed when they first hit $b$.