Recent success in manipulating ultracold atomic systems allows to probe different strongly correlated regimes in one dimension. I will explain how, experimentally, 1D tubes are defined by turning on a 2D optical lattice. For repulsive interactions, regimes such as the spin-coherent Luttinger liquid and the spin-incoherent Luttinger liquid can be realized by tuning the inter-atomic interaction strength and trap parameters. Due to the trap potential the density decreases near the edges of the tubes and the spin-incoherent regime is inevitably realized. In general, the spin-coherent Luttinger liquid regime in the center of the tube crosses over to its spin-incoherent counterpart at the edges. We identify the noise correlations of density fluctuations as a robust observable to discriminate between these two regimes. On the other hand, for attractive interactions, we study pairing in spin-imbalanced ultracold atomic system of fermions in 1D to identify exotic states such as the 1D analog of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. Calculations are done using the Bethe Ansatz technique and the trap is incorporated into the solution via a local density approximation. The Thermodynamic-Bethe-Ansatz equations were solved numerically and different local density profiles were calculated in the trap for different finite temperatures. A scheme to identify the phase diagram using local density profiles in the trap is proposed that would be immediately useful for experimentalists. Finally, I will summarize and address the concrete prospects for realizing and probing these phenomena experimentally using optical lattices.