  In this review paper, we treat the topic of fine gradings of Lie algebras.
This concept is important not only for investigating the structural properties
of the algebras, but, on top of that, the fine gradings are often used as the
starting point for studying graded contractions or deformations of the
algebras. One basic question tackled in the work is the relation between the
terms 'grading' and 'group grading'. Although these terms have originally been
claimed to coincide for simple Lie algebras, it was revealed later that the
proof of this assertion was incorrect. Therefore, the crucial statements about
one-to-one correspondence between fine gradings and MAD-groups had to be
revised and re-formulated for fine group gradings instead. However, there is
still a hypothesis that the terms 'grading' and 'group grading' coincide for
simple complex Lie algebras. We use the MAD-groups as the main tool for finding
fine group gradings of the complex Lie algebras $A_3 \cong D_3$, $B_2 \cong
C_2$, and $D_2$. Besides, we develop also other methods for finding the fine
(group) gradings. They are useful especially for the real forms of the complex
algebras, on which they deliver richer results than the MAD-groups. Systematic
use is made of the faithful representations of the three Lie algebras by
$4\times 4$ matrices: $A_3 = sl(4,\mathbb C)$, $C_2 = sp(4,\mathbb C)$, $D_2 =
o(4,\mathbb C)$. The inclusions $sl(4,\mathbb C)\supset sp(4,\mathbb C)$ and
$sl(4,\mathbb C) \supset o(4,\mathbb C)$ are important in our presentation,
since they allow to employ one of the methods which considerably simplifies the
calculations when finding the fine group gradings of the subalgebras
$sp(4,\mathbb C)$ and $o(4,\mathbb C)$.
