The aim of this paper is to show that Euler’s exponential formula $\lim_{n\rightarrow\infty}\linebreak[4] (I-tA/n)^{-n}x = e^{tA}x$, well known for $C_0$ semigroups in a Banach space $X\ni x$, can be used for semigroups not of class $C_0$, the sense of the convergence being related to the regularity of the semigroup for $t>0$. Although the strong convergence does not hold in general for not strongly continuous semigroups, an integrated version is stated for once integrated semigroups. Furthermore by replacing the initial topology on $X$ by some (coarser) locally convex topology $\tau$, the strong $\tau$-convergence takes place provided the semigroup is strongly $\tau$-continuous; in particular this applies to the class of bi-continuous semigroups. More generally if a $k$-times integrated semigroup $S(t)$ in a Banach space $X$ is strongly $k$-times $\tau$-differentiable, then Euler’s formula holds in this topology with limit $S^{(k)}(t)$. On the other hand, for bounded holomorphic semigroups not necessarily of class $C_0$, Euler’s formula is shown to hold in operator norm, with the error bound estimate ${\cal O}(\ln n/n)$, uniformly in $t>0$. All these results also concern degenerate semigroups.