  We consider records and sequences of records drawn from discrete time series
of the form $X_{n}=Y_{n}+cn$, where the $Y_{n}$ are independent and identically
distributed random variables and $c$ is a constant drift. For very small and
very large drift velocities, we investigate the asymptotic behavior of the
probability $p_n(c)$ of a record occurring in the $n$th step and the
probability $P_N(c)$ that all $N$ entries are records, i.e. that $X_1 < X_2 <
... < X_N$. Our work is motivated by the analysis of temperature time series in
climatology, and by the study of mutational pathways in evolutionary biology.
