Euler's beautiful formula $\zeta(2n) = -(2\pi i)^{2n}/(2(2n)!)B_{2n}$
can be seen as
the starting point of the investigation of special values of
L-functions. In particular, Euler's result shows that all critical zeta
values are rational up to multiplication with a particular period, here
the period is a power of $(2\pi i)$. Conjecturally this is expected to hold
for all critical L-values of motives. In this talk, we will focus on
L-functions of number fields. In the first part of the talk, we will
discuss the 'critical' and 'non-critical' L-values exemplary for
the Riemann zeta function. Afterwards, we will head on to more general
number fields and explain our recent joint result with Guido Kings on
the algebraicity of critical Hecke L-values for totally imaginary fields
up to explicit periods.