A block design setting is an integer triple (v,b,k) specifying the numbers of treatments, blocks, and experimental units per block, respectively, available for an experiment. If v k, i.e. if it is not feasible to have a single replication of every treatment in each block, then (v,b,k) is called an incomplete block design setting. Consider comparing the relative effectiveness of v tretments employing n=bk experimental units arranged in to b blocks of size k. The standard linear model for the observation y ju ، on unit u in block j when using designed is, y_{ju}=\\mu+\au_{d[j,u]}+\\beta_j+\\varepsilon_{ju} \\quad j=1,2,...,b \\quad u=1,2,...,k .\\\\ The components of model are an overall mean \\mu , the effect $\au_{d[j,u]} $ of treatment $d[j,u] $ assigned to unit $u $ in block $ j$ by design $ d$, a block effect $\\beta_j $, and a random error $\\varepsilon_{ju} $ with zero mean . Writing $\\boldsymbol{\au}=(\au_1,\au_2,...,\au_v)'$ and $\\boldsymbol{\\beta}=(\\beta_1,\\beta_2,...,\\beta_b)'$ for the vectors of treatment and block effects and $\extbf{y}=(y_1,y_2,...,y_n)'$ for the responses vector, then mean vector is $E(\\boldsymbol{y})=\\mu\\mathbf{1}_n+M_d \\boldsymbol{\au}+L\\boldsymbol{\\beta}$ for $L=I_b\\otimes \\mathbf{1}_k$ the $n\imes b$ unit $ / $ block design matrix for blocks, and $M_d$ the $n\imes v$ unit $ / $treatment design matrix for treatments.It is well known that all treatment contrasts are estimble under $ d$ if and only if the information matrix for estimation of treatment effects , $ C_d=M'_d(I-k^{-1}LL')M_d $ , is of rank $v-1 $ ; equivalently, the rank of $X_d=(\\mathbf{1}_n \\mid M_d \\mid L)$ is $b+v-1 $. A design with this property is said to be connected. A necessary condition for $d $ to be connected is that the number of row s of $X_d $ is at least its required rank, that is, $ n \\geq b+v-1 $. Let $D(v,b,k)$ denote the $bk=b+v-1$ \\\\ The $D(v,b,k)$ is the Since optimality criteria are different and usually have different results , the best designis a design is a design that is knownto enjoy many optimalities. In this thesis we study the basic properties of minimally connected block designs, and some of optimality criteria such as M, A, MV, D and E that have been studied in the previous research.When observation are uncorrelated the A-, MV-, D- and optimal designs in $ (v,b,k) $ are known, all connected block designs are D- equal. In this thesis optimality of designs in $ (v,b,k) $ is studied when observations are correlated. For the D-optimality problem an arbitrary correlation structure is considered. Thus, all connected block designs are shown to be D- equal regardless of the correlation structure. For other opitimality problems this spatial correlation structure is assumed:\\\\ \\begin{equation*} Cov(y_{j,u},y_{j',u'})=\\left \\{ \\begin{array}{cc} 0 j\eq j' \\\\ \\sigma^2 j=j' , u=u' \\\\ \\rho_{|u-u'|}\\sigma^2 j=j' , u\eq u'\\\\ \\end{array} \\end{equation*} where \\begin{equation*} 1 \\rho_1 \\geq \\rho_2 \\geq \\rho_3 \\geq...\\geq \\rho_{k-1} \\geq 0. \\end{equation*} In addition it is every where required that the variance-covariance matrix \\Sigma for the entire $n\imes 1$ observations vector $y $ be positive definite, i.e.\\\\ $\\boldsymbol{a}'{\\Sigma}\\boldsymbol{a} \\geq 0$ for any $\\boldsymbol{a} \eq 0 $.\\\\ M-optimal (this includes A- and MV-optimal) designs for estimation of elementary treatment contrasts are identified in the thus we submit new results for M- and E-optimal designs.We attempt to constructe the best possible design for each of criteria.