  Sturmian sequences are well-known as the ones having minimal complexity over
a 2-letter alphabet. They are also the balanced sequences over a 2-letter
alphabet and the sequences describing discrete lines. They are famous and have
been extensively studied since the 18th century. One of the {extensions} of
these sequences over a $k$-letter alphabet, with $k\geq 3$, are the episturmian
sequences, which generalizes a construction of Sturmian sequences using the
palindromic closure operation. There exists a finite version of the Sturmian
sequences called the Christoffel words. They are known since the works of
Christoffel and have interested many mathematicians. In this paper, we
introduce a generalization of Christoffel words for an alphabet with 3 letters
or more, using the episturmian morphisms. We call them the {\it epichristoffel
words}. We define this new class of finite words and show how some of the
properties of the Christoffel words can be generalized naturally or not for
this class.
