  In this paper we study the drifted Brownian meander, that is a Brownian
motion starting from $ u $ and subject to the condition that $ \min_{ 0\leq z
\leq t} B(z)> v $ with $ u > v $. The limiting process for $ u \downarrow v $
is analyzed and the sufficient conditions for its construction are given. We
also study the distribution of the maximum of the meander with drift and the
related first-passage times. The representation of the meander endowed with a
drift is provided and extends the well-known result of the driftless case. The
last part concerns the drifted excursion process the distribution of which
coincides with the driftless case.
