In [1] was shown the existence of the functionals of the kind of Banach limits defined on the real vector space $\mathbf{m}$ of all bounded sequences in a real normed space $X$. In [2] by these functionals was defined the almost convergence of a sequence $(x_{i})\in \mathbf{m}$ and shown that $(x_{i})$ almost converges to $s\in X$ iff \[\biggl\| \frac{1}{p}um_{i=0}^{p-1} X_{k+i} - s\biggr\|_{X} o 0\quadext{as } po ıfty\] uniformly in $k=(0,1,2,\dots)$. 2. The basic idea in this paper is to obtain a new method of sum ability of the vector sequences $(x_{i})\in \mathbf{m}$. The paper is organized as follows. First, we will show the existence of an another family of functionals (of the kind of Banach limits) defined on the space $\mathbf{m}$. Next, we define, by these functionals, a new method of sum ability of sequences $(x_{i})\in \mathbf{m}$ which will be called quasi almost convergence. Further, we will show a theorem which contains a necessary and sufficient condition for a sequence $(x_{i})\in \mathbf{m}$ to be quasi almost convergent. Next, we shall prove two theorems which shows that the class of quasi almost convergent sequences lies between the class of almost convergent sequences and the class of $C$-summable sequences.