We will see the Hirzebruch-Riemann-Roch theorem as the generalization of classical Riemann-Roch theorem for Riemann Surface.
For a compact complex manifold $X$ and the complex vector bundle $\pi:E\longrightarrow X$,
$$
\chi\left(X, E\right)= \int_{M}{\rm ch}\left(E\right){\rm Td}\left(E\right)
$$
where $\chi\left(X, E\right)$, ${\rm ch}\left(E\right)$, and ${\rm Td}\left(E\right)$ are Euler characteristics, Chern Character, and Todd class.