This thesis discusses graph marking game and graph colouring game. Suppose G=(V, E) is a graph. A marking game on G is played by two players, Alice and Bob, with Alice playing first. At the start of the game all vertices are unmarked. A play by either player consists of marking an unmarked vertex. The game ends when all vertices are marked. For each vertex v of G, write t(v)=j if v is marked at the jth step. Let s(v) denote the number of neighbours u of v for which t(u) < t(v), i.e., u is marked before v. The score of the game is $$s = 1+ max_{v in V} s(v).$$ Alice's goal is to minimize the score, while Bob's goal is to maximize it. The game colouring number colg(G) of G is the least s such that Alice has a strategy that results in a score at most s. Suppose r geq 1, d geq 0 are integers. In an (r, d)-relaxed colouring game of G, two players, Alice and Bob, take turns colouring the vertices of G with colours from a set X of r colours, with Alice having the first move. A colour i is legal for an uncoloured vertex x (at a certain step) if after colouring x with colour i, the subgraph induced by vertices of colour i has maximum degree at most d. Each player can only colour an uncoloured vertex with a legal colour. Alice's goal is to have all the vertices coloured, and Bob's goal is the opposite: to have an uncoloured vertex without legal colour. The d-relaxed game chromatic number of a graph G, denoted by $chi_g^{(d)}(G)$ is the least number r so that when playing the (r, d)-relaxed colouring game on G, Alice has a winning strategy. If d=0, then the parameter is called the game chromatic number of G and is also denoted by $chi_g(G)$. This thesis obtains upper and lower bounds for the game colouring number and relaxed game chromatic number of various classes of graphs. Let colg(PT_k) and colg(P) denote the maximum game colouring number of partial k trees and the maximum game colouring number of planar graphs, respectively. In this thesis, we prove that colg(PT_k) = 3k+2 and colg(P) geq 11. We also prove that the game colouring number colg(G) of a graph is a monotone parameter, i.e., if H is a subgraph of G, then colg(H) leq colg(G). For relaxed game chromatic number of graphs, this thesis proves that if G is an outerplanar graph, then $chi_g^{(d)}(G) leq 7-t$ for $t= 2, 3, 4$, for $d geq t$, and $chi_g^{(d)}(G) leq 2$ for $d geq 6$. In particular, the maximum $4$-relaxed game chromatic number of outerplanar graphs is equal to $3$. If $G$ is a tree then $chi_ g^{(d)}(G) leq 2$ for $d geq 2$.