  Let $f:A \to B$ be a ring homomorphism and $J$ an ideal of $B$. In this
paper, we initiate a systematic study of a new ring construction called the
"amalgamation of $A$ with $B$ along $J$ with respect to $f$". This construction
finds its roots in a paper by J.L. Dorroh appeared in 1932 and provides a
general frame for studying the amalgamated duplication of a ring along an
ideal, introduced and studied by D'Anna and Fontana in 2007, and other
classical constructions such as the $A+ XB[X]$ and $A+ XB[[X]]$ constructions,
the CPI-extensions of Boisen and Sheldon, the $D+M$ constructions and the
Nagata's idealization.
