We present it powerful connectivity-altering Monte Carlo (MC) algorithm for simulating atomistically detailed models of long- and short-chain branched polymers. Based on a mix of advanced and simple moves, the algorithm allows the robust simulation of chain systems with a variety of molecular architectures: H-shaped, A(3)AA(3) multiarm (pom-pom), and short-chain branched (SCB) ones. For the H-shaped and A(3)AA(3) architectures (A denotes the backbone and A the arm), in particular, the recently developed intermolecular double bridging (DB) move effecting a new bridging between the chain backbones or the branches of two different chains and the intramolecular double rebridging (IDR) move effecting it new bridging between two branches of the same chain [Karayiannis et al. J. Chem. Phys. 2003, 118, 2451] are shown to provide it robust Sampling of their structural and confomational characteristics at the chain level. The double concerted rotation (d-CONROT) and H-shaped branch rebridging (H-BR) moves, on the other hand, which are important in relaxing internal degrees of freedom in the chain and the local structure around branch points, allow complete equilibration at the segmental level, which is prerequisite for the correct prediction of the volumetric and packing properties of the simulated systems. For the case of SCB polymers, a continuum configurational bias move for branched polymers (Br-CCB) has been designed capable of sampling changes in the atomic coordinates in the vicinity of branch points. Along with an effective implementation of the conventional end-bridping (EB) move, this has resulted in a powerful method for simulating SCB polymers with it prespecified distribution of branch points along the chain and given (i.e., fixed) number of carbon atoms per branch. To deal with polydispersity in the simulated systems, the algorithm is executed in a semigrand canonical ensemble by making use of the set of chain relative chemical potentials which for it bulk, linear system produces it uniform distribution of chain lengths [Pam and Theodorou Macromolecules 1995, 28, 7224]. Consistent with recent considerations by Ramos et al. [Macromolecules 2007, 40, 9640], two different chain length distributions should be simultaneously controlled in the course of the simulation for branched systems. The new, generalized MC algorithm is capable of thoroughly equilibrating model H-shaped, multiarm A(3)AA(3) and SCB polymers at all length scales and adequately sampling fluctuations in their structural, volumetric, and conformational properties. overall, we find that branched polymers are characterized by it stiffer conformation than linear analogues of the same total chain length. For the H-shaped and A(3)AA(3) architectures, in particular, we have observed that each dangling arm exerts an entropically driven tensile force on the chain crossbar (the part of the chain between the two branch points), causing significant chain stretching. Our results extend the previous works of Karayiannis et al. [J. Chem. Phys. 2003, 118, 2451] on the simulation of H-shaped polymers to A(3)AA(3) PE polymers and of Ramos et al. [Macromolecules 2007, 40, 9640] on the simulation of model ethylene-1-hexene copolymers to copolymers with longer alpha-olefinic side groups, such as file ethylene-1-octene and ethylene-1-decene ones.