Considering polynomial sets with complex coefficients \[ \{P_n\}_{n\geq 0}\;\text{ with deg}\,P_n=n, \] the authors study the problem of determining the solution to the Connection problem: given two polynomial sets \(\{P_n\}_{n\geq 0},\;\{Q_n\}_{n\geq 0}\), find the coefficients in \(Q_n(x)=\sum_{m=0}^n\, c_m(n)P_m(x)\). The problem is not studied in its full generality of course: the authors restrict themselves to Boas-Buck polynomial sets. These polynomials are given by a generating function of Boas-Buck type: \[ \sum_{n=0}^{\infty}\,\lambda_nP_n(x)t^n=A(t)B(xC(t)), \] with constants \(\{\lambda_n\}\) and where \[ A,B\text{ are power series }\sum_{n=0}^{\infty}\,a_nt^n,\;a_0\not= 0,\text{ resp. }\sum_{n=0}^{\infty}\,b_nt^n,\;b_0\not= 0 \] and \[ C\text{ is a power series }\sum_{n=1}^{\infty}\,c_nt^n,\;c_1\not= 0. \] Denote by \(C^{\ast}\) the inverse power series of \(C\) with respect to composition: \[ C^{\ast}(C(t))\equiv t,\quad C(C^{\ast}(s))\equiv s, \] then the main result is Theorem. Let \(\{P_n\}_{n\geq 0},\;\{Q_n\}_{n\geq 0}\) be two Boas-Buck polynomial sets generated by \[ A_1(t)B_1(xC_1(t))=\sum_{n=0}^{\infty}\,{P_n(x)\over n!}\,t^n,\quad A_2(t)B_2(xC_2(t))=\sum_{n=0}^{\infty}\,{Q_n(x)\over n!}\,t^n, \] then the connection coefficients are given by \[ c_m(n,a)={n!\over m!}\,\sum_{k=m}^n\, a_k(n) b_m(k) a^k\, {\gamma_k^{(2)}\over\gamma_k^{(1)}}, \] where \[ \begin{aligned} A_2(t)\left(C_2(t)\right)^m&= \sum_{n=m}^{\infty}\,a_m(n)t^n, \\ {\left(C_1^{\ast}(t)\right)^m\over A_1(C_1^{\ast}(t))}&= \sum_{n=m}^{\infty}\,b_m(n)t^n,\\ \text{and} B_i(t)&= \sum_{k=0}^{\infty}\,\gamma_k^{(i)} t^k\;(i=1,2). \end{aligned} \] After this main result follows a number of interesting corollaries pertaining to different types of polynomial sets (Brenke-, Panda-, and Humbert-polynomial sets) and applications to Brafman-, Laguerre-, Koekoek-Laguerre-Sobolev-, Chaundry-, Srivastava-Pathan-, Wilson-, Racah-, Jaco\-bi-, generalized Rice-, generalized Bessel-, Srivastava-, modified Jacobi- and modified Laguerre-polynomials. The paper concludes with a section on the connection coefficients for the classical continuous and classical discrete sets of orthogonal polynomials in terms of \(x^n\) (continuous case) resp. \((-x)_n=(-x)(-x+1)\cdots (-x+n-1)\) (discrete case) and each of the continuous (discrete) polynomials in each of the other continuous (discrete) orthogonal polynomials.