  We critically examine the current status of theoretical calculations of the
energies, the fine structure, and the isotope shift of the lowest-lying states
of helium, searching for unresolved discrepancies with experiments.
Calculations are performed within the quantum electrodynamics expansion in
powers of the fine structure constant $\alpha$ and the electron-to-nucleus mass
ratio $m/M$. For energies, theoretical results are complete through orders
$\alpha^6m$ and $\alpha^6m^2/M$, with the resulting accuracy ranging from $0.5$
to $2$~MHz for the $n=2$ states. The fine-structure splitting of the $2^3P$
state is predicted with a much better accuracy, 1.7~kHz, as a consequence of a
calculation of the next-order $\alpha^7m$ effect. An excellent agreement of the
theoretical predictions with the recent measurements of the fine structure
provides one of the best tests of the bound-state QED in few-electron systems
and determines the fine-structure constant $\alpha$ with an accuracy of 31~ppb.
The isotope shift between $^3$He and $^4$He is treated theoretically with a
sub-kHz accuracy, which allows for a high-precision determination of the
differences of the nuclear charge radii $\delta r^2$. Several such
determinations, however, yield results that are in a 4$\sigma$ disagreement
with each other, what remains unexplained. Apart from this, we find no
significant discrepancies between theory and experiment for the helium atom. In
the future, a calculation of the next-order $\alpha^7m$ effect for energy
levels will enable determinations of the nuclear charge radii from atomic
transition frequencies with an accuracy better than 1\%. Combined with the
complementary determinations from muonic atoms, this will provide a sensitive
test of universality in electromagnetic interactions of leptons and contribute
to the solution of the proton charge radius puzzle.
