This paper is devoted to the coincidence theory of two continuous mappings. A definition is given, in cohomological terms, of the coincidence index $I_{f,g}$ of two continuous mappings $f,g\colon M\to N$, where $M$ and $N$ are connected (not necessarily compact), orientable, $n$-dimensional topological manifolds without boundary, $f$ is a compact mapping and $g$ is a proper mapping. Invariance of the index $I_{f,g}$ under compact homotopies of $f$ and proper homotopies of $g$ is proved. It is shown that $I_{f,g}\ne0$ is a sufficient condition for the existence of coincidence points of $f$ and $g$. The Lefschetz number $\Lambda_{f,g}$ for $f$ and $g$ is also defined. The main result of the paper is a theorem on the coincidence of the numbers $\Lambda_{f,g}$ and $I_{f,g}$. Bibliography: 7 titles.