The matching number of a family of subsets of an $n$-element set is the maximum number of pairwise disjoint sets. The families with matching number $1$ are called intersecting. The famous Erd\H os-Ko-Rado theorem determines the size of the largest intersecting family of $k$-sets. Its generalization to the families with larger matching numbers, known under the name of the Erdős Matching Conjecture, is still open for a wide range of parameters. In this paper, we address the degree versions of both theorems. More precisely, we give degree and $t$-degree versions of the Erdős-Ko-Rado and the Hilton-Milner theorems, extending the results of Huang and Zhao, and Frankl, Han, Huang and Zhao. We also extend the range in which the degree version of the Erdős Matching conjecture holds.