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The search for novel phases in strongly correlated systems is one of the most active frontiers in contemporary condensed matter physics. In this thesis the author explores the ground state phases of correlated electrons in one-dimension, modelled by the extended Hubbard model at half-filling, focusing on the novel bond-order-wave phase at small to intermediate couplings - a previously unknown phase whose existence was conclusively established by the author and his co-workers. The second problem studied involves the Peierls transition in correlated electrons interacting with finite frequency phonons.Both these topics have direct relevance to modelling and thus gaining an understanding of the properties of real quasi one-dimensional materials. An important aspect of the thesis is the in-depth exposition of the Stochastic Series Expansion (SSE) - a quantum Monte Carlo method that has, in the recent years, evolved into one of the most powerful computational techniques to study strongly correlated systems. The author has contributed to the development of the method and as such has a unique perspective which is bound to benefit anyone planning to learn SSE.