Let $F_{g,r}$ be a compact, oriented surface of genus $g$ with $r\ge0$ boundary components. The mapping class group $\Gamma_{g,r}$ of $F$ is the path-components of the group of orientation-preserving diffeomorphisms of $F$ fixing the boundary pointwise.

The first part of this thesis concerns homology stability of the mapping class group. The map gluing a pair of pants onto $F$ along one or two boundary components induce inclusions of the corresponding mapping class groups. Homology stability is the statement that these inclusions induce isomorphism on homology when the genus $g$ is large enough compared to the homology degree $n$. In this paper we prove homology stability for surfaces with boundary with integral coefficients in a range that is at most one homology degree away from the best possible range. We extend this near-optimal result to twisted coefficients, i.e. coefficients with an action of the mapping class group.

The second, short part of the thesis compares different versions of mapping class groups of a surface, namely the groups of diffeomorphisms, homeomorphisms, and homotopy equivalences, respectively. It is a classical result that the path-components of these groups are isomorphic, and we give a short, self-contained, elementary proof, which has been missing from the literature.