  Methylammonium lead triiodine (MAPbI$_3$) is a material representative of the
hybrid organic-inorganic lead halide perovskites which attract currently great
attention due to their photovoltaic efficiency and bright optoelectronic
properties. Here, the coherent spin dynamics of charge carriers and spin
dependent phenomena induced by the carrier interaction with nuclear spins are
studied in MAPbI$_3$ single crystals, using time-resolved Kerr rotation at
cryogenic temperatures in magnetic fields up to 3 T. Spin dephasing times up to
a few nanoseconds and a longitudinal spin relaxation time of 37 ns are
measured. The Larmor spin precession of both resident electrons and holes is
identified in the Kerr rotation signals. The Land\'e factors ($g$-factors) in
the orthorhombic crystal phase show a strong anisotropy, ranging for the holes
from $-0.28$ to $-0.71$ and for the electrons from $+2.46$ to $+2.98$, while
the $g$-factor dispersion of about 1% is rather small. An exciton $g$-factor of
$+2.3$ is measured by magneto-reflectivity. A dynamic nuclear polarization by
means of spin polarized electrons and holes is achieved in tilted magnetic
fields giving access to the carrier-nuclei exchange interaction and the nuclei
spin relaxation time exceeding 16 minutes.
