Let \(B=\bigoplus_{\nu\in \mathbb N^r} [B]_\nu\) be a standard \(\mathbb N^r\)-graded algebra over an Artin local ring \(A=[B]_{(0,0,\dots,0)}.\) Let \(M=\bigoplus_{\nu\in \mathbb Z^r}[M]_\nu\) be a finite \(\mathbb Z^r\)-graded \(B\)-module. Let \(\ell\) denote length. The Hilbert function of \(M\) is defined as \(H_M(\nu)=\ell_A([M]_\nu)\) for all \(\nu\in \mathbb Z^r.\) For all large \(\nu\), \(H_M(\nu)\) is given by a polynomial \(P_M(\mathbf{X})=P_M(X_1, X_2, \ldots, X_r)\) with rational coefficients. Let \(\mathcal N\) denote the multigraded irrelevant ideal \(\mathcal N=\bigoplus_{v_1, v_2,\ldots, v_r> 0}[B]_\nu.\) The relevant support of \(M\) is defined as \(\text{Supp}_{++}(M)=\{{\mathfrak p}\in \text{Supp} (M)\mid {\mathfrak p} \text{ is } {\mathbb N^r-}\text{graded and } {\mathcal N} \not\subset {\mathfrak p}\}.\) Then \(\deg P_M(\mathbf{X})=d_{++}:=\dim \text{Supp} _{++}(M).\) Write \(P_M(\mathbf{X})\) as \[P_M(\mathbf{X})=\sum_{n_1, n_2, \ldots, n_r\geq 0}e(n_1, n_2, \ldots, n_r)\binom{X_1+n_1}{n_1}\dots \binom{X_r+n_r}{n_r}.\] If \(|\mathbf{n}|=n_1+n_2+\dots+n_r=d_{++}\) then \(e(n_1, n_2, \ldots, n_r):=e(\mathbf{n}, M)\) is a non-negative integer called the mixed multiplicity of \(M\) of the type \((n_1, n_2, \ldots, n_r).\) Mixed multiplicities are defined using the Hilbert series of \(M\) in this paper. let \(d=\dim M.\) Define the Hilbert series of \(M\) by \(\text{Hilb}_M(t_1, t_2, \ldots, t_r)=\sum_{\nu\in \mathbb Z^r}\ell_A([M]_\nu)t_1^{\nu_1}t_2^{\nu_2}\ldots t_r^{\nu_r}.\) It is proved that there are unique Laurent polynomials \(Q_{\mathbf{n}}(\mathbf{t})\) so that \(\text{Hilb}_M(\mathbf{t})=\sum_{|\mathbf{n}|=d}\frac{Q_{\mathbf{n}} (\mathbf{t})} {(1-t_1)^{n_1}(1-t_2)^{n_2}\ldots(1-t_r)^{n_r}}.\) Put \(\mathbf{1}=(1,\ldots, 1)\) and \(\mathbf{n}=(n_1, \ldots, n_r)\in \mathbb Z^r\) and \(|\mathbf{n}|=d-r.\) The mixed multiplicity of \(M\) of the type \(\mathbf{n}\) is defined as \(e_{\mathbf{n}}(M)=Q_{\mathbf{n}+1}(\mathbf{1}).\) The two notions of the mixed multiplicities are related by the equation \(e(\mathbf{n}, M)=e_{\mathbf{n}}(M/H_{\mathcal N}(M))\) where \(\mathbf{n}=d_{++}.\) Basic properties of mixed multiplicities are proved using filter-regular elements. The main objective of the paper is to use the mixed multiplicities to find the classical invariants in algebraic geometry called projective degrees of rational maps. Let \(k\) be a field and \(R=k[x_0, x_1, \ldots, x_d]\). Let \(n \geq d\) and \(\mathcal F : \mathbb P^d_k=\text{Proj}(R) \xrightarrow \;\mathbb P^n_k\) be a rational map defined by \(n+1\) homogeneous polynomials \(f_0, f_1, \ldots, f_n\in R\) of degree \(\delta>0\) and let \(I=(f_0, f_1, \ldots, f_n).\) The projective degrees of \(\mathcal F,\) denoted by \(d_i(\mathcal F),\) are defined to be the multidegrees of the graph of \(\mathcal F.\) The projective degrees are calculated using the special fiber rings. Using these techniques, the projective degrees of rational maps defined by perfect height two ideals and Gorenstein height three ideals are computed. The formulas for \(d_i(\mathcal F)\) are given in terms of the free resolutions of these ideals.