A proper vertex coloring of graph G , is an assignment of colors to the vertices of G such that, no two adjacent vertices receive the same color. A list assignment L for a graph G assigns to each vertex v ? V ( G ) a set L ( v ) of allowable colors. An L -coloring of G is a proper vertex coloring such that for every v 2 V ( G ) the color on v belongs to L ( v ). For example, when colors represent time periods and vertices are jobs, the list model incorporates restrictions that not all time periods are suitable for all jobs. A list assignment L for G is k -uniform if j L ( v ) j = k for all v 2 V ( G ). A proper vertex coloring of a graph is m -bounded if each color appears on at most m vertices. Given a k -uniform list assignment L for an n -vertex graph G , we say that G is equitably L -colorable if G has an ? n/k ?-bounded L coloring. A graph G is equitably list k -colorable or equitably k -choosable if G is equitably L -colorable whenever L is a k -uniform list assignment for G . Kostochka, et al. 2003, introduced equitable list coloring. Kostochka et al. 2003, proved that, if G is a graph and k ? max { ?( G ) , n ( G ), then G is equitably k -choosable unless G contains K k +1 or is K k;k (with k odd in the latter case). They expressed the following conjecture. Conjecture: Every graph G is equitably k -choosable, whenever k ?( G ). Kostochka et al. 2013, proved that this conjecture yields for ? ? 7 and show that every graph with maximum degree at most r and at least r 3 vertices is equitably ( r +2)-choosable. These results are studied in Chapter 2. Kostochka et al. 2003, proved that a graph G is equitably k -choosable if G is a connected interval graph and k ? ?( G ), or G is a 2-degenerate graph and k ? max { ?( G ) ; 5}. Pelsmajer 2003, proved that every graph G is equitably k -choosable for any k _ ?( G )(?( G ) - 1)/2+ 2. He proved that this conjecture yields for ? ? 3. These results are studied in Chapter 3. Pelsmajer 2008, proved this conjecture for graphs of treewidth w ? 5 if also k ? 3 w - 1. He also show that if G has treewidth w ? 5, then G is equitably k -choosable for k ? max { ?( G ), - 3 w - 1 } . As a corollary, if G is chordal, then G is equitably k -choosable for k ? 3?( G ) - 4 when ?( G ) 2. Xin Zhang and Jian-Liang Wu 2011, proved that every series–parallel graph with maximum degree ? is equitably k -choosable whenever k ? max { ?( G ) ; 3}. These results are studied in Chapter 4. Qiong Li and Yuehua Bu 2009, proved this Conjecture for planar graphs with ?( G ) ? 6 and no 4 or 6-cycles. Junlei Zhu and Yuehua Bu 2010, proved that outerplaner graphs are equitably k -choosable, whenever k ? ?( G ). Aijun Dong et al. 2012, proved that if G is a planar graph without 5 and 7-cycles, then G is equitably k -choosable and equitably k -colorable, where k _ max f ?( G ) , 7 } . Aijun Dong et al. 2013, proved that every planar graph G without 4-cycles is equitably k -choosable, where k _ max { ?( G ) , 7 } . Junlei Zhu and Yuehua Bu 2015, proved that every planar graph G is equitably k -choosable if one of the following conditions holds: (1) G is triangle-free and k _ max f ?( G ) ; 8 g . (2) G has no 4- and 5-cycles and k ? max { ?( G ) , 7 } . These results are studied in Chapter 5.