A (finite) graph is odd if all its vertices have odd degrees. The principal aim of this survey is to present the current state of research on covers and decompositions of graphs into fewest possible number of odd subgraphs. Given a graph $G$, the parameters ${\chi }_o^{\prime }\left(G\right)$ and ${\mathrm{cov}}_o\left(G\right)$ denote, respectively, the minimum size of a decomposition and cover of $G$ consisting of odd subgraphs. Pyber (1991) and Mátrai (2006), respectively, have shown that for every simple graph $G$ it holds that ${\chi }_o^{\prime }\left(G\right)\le 4$ and ${\mathrm{cov}}_o\left(G\right)\le 3$, with both bounds being sharp. The multigraph analogues of the same inequalities, given by the present authors in 2018 and 2019, respectively, are discussed in detail. The list versions of the graph parameters ${\chi }_o^{\prime }\left(G\right)$ and ${\mathrm{cov}}_o\left(G\right)$, along with other generalizations and possible new directions of related research, are considered in the latter part of the article. Throughout we also pose various structural and algorithmic questions and problems.

如果一个（有限）图的所有顶点都具有奇数度，则它是奇数。这项调查的主要目的是介绍图的覆盖和分解到最少可能的奇数子图数量的研究现状。给定图$G$，参数 ${\chi }_{Ø}^{\prime }（G）$ 和 ${\mathrm{冠状病毒}}_{Ø}（G）$ 分别表示分解的最小大小和 $G$由奇数个子图组成。Pyber（1991）和Mátrai（2006）分别表明，对于每个简单图$G$ 它认为 ${\chi }_{Ø}^{\prime }（G）\le 4$ 和 ${\mathrm{冠状病毒}}_{Ø}（G）\le 3$，两个边界都非常清晰。本作者分别于2018年和2019年给出的具有相同不等式的多图类似物进行了详细讨论。图形参数的列表版本${\chi }_{Ø}^{\prime }（G）$ 和 ${\mathrm{冠状病毒}}_{Ø}（G）$在本文的后半部分中，我们将结合其他概括和相关研究的可能新方向进行研究。在整个过程中，我们还提出了各种结构和算法问题。