This dissertation is primarily focused on conditions for the existence of spanning closed trails in graphs. However, results in my dissertation and the method we used, which was invented by Catlin, are not only useful for finding spanning closed trails in graphs, but also useful to study double cycle cover problems, hamiltonian line graphs problems, and dominating closed trail problems, etc. A graph is called supereulerian if it contains a spanning closed trail. Several, people have worked on the conditions for spanning closed trails having the form "d(u) + d(v) $>$ cn (0 $<$ c $<$ 1)" for all edge uv in graph, as in the paper of Benhocine, Clark, Kohler and Veldman (3), etc. When obtaining good sufficient conditions having the form "d(u) + d(v) $>$ cn", it is useful to show that a graph G is either supereulerian or it can be contracted to a nonsupereulerian graph G$\sp\prime$ having a large matching. This was done. For example, we show that if a 3-edge-connected simple graph has no spanning closed trail then it can be contracted to a nonsupereulerian graph G$\sp\prime$ of order n $\sp\prime$ whose maximum matching has size at least (n$\sp\prime$ + 4)/3, and it is best possible. By using this result, we show that if G is a 3-edge-connected simple graph of order n, if for every edge uv, d(u) + d(v) $\geq$ $n \over 5$ $-$ 2, then either G has a spanning closed trail of G is contractible to the Petersen graph. By a theorem of Harary and Nash-Williams, this implies that either the line graph L(G) is hamiltonian or G can be contracted to the Petersen graph. This proves a conjecture of Benhocine, Clark, Kohler and Veldmann (3) for 3-edge-connected graphs, and with a stronger conclusion. We also study the conditions for supereulerian graphs having form "$\sum\sbsp{i=1}{t} d(u\sb{i})$ $>$ cn" (0 $<$ c $<$ 1) where t is a positive integer and no two vertices of $\{ u\sb1,\ u\sb2,\cdots u\sb{t} \}$ are adjacent. We obtain some best possible results on this aspect which improve the results of Benhocine, Clark, Kohler, & Veldman (3), Calin (11), (13), (14), Clark (24), Z. Q. Chen & Y. F. Xue (23), Lesniak-Foster & Willianson (30) and Veldman (36) significantly. We also study the following extremal graph theory problem: For a family F of graphs and for a natural number n, what is the maximum size of simple graphs of order n which are not in F, where F = $\{$supereulerian graphs with clique number $m\}$. (Note that when F = $\{$graphs with clique number at least $m\}$, this is Turan's Theorem.) One of our results proves a conjecture of Cai (10).