We review some recent theoretical and experimental developments of one-dimensional few-body problems in ultracold atomic system. The experiments have so far realized the deterministic loading of few atoms in the ground state of a potential well, the observation of tunneling dynamics out of the metastable trap controlled by a magnetic gradient for a repulsively or attractively interacting system, the preparation of two fermionic atoms in an isolated double-well potential with a full control over the quantum state of the system, the formation of a Fermi sea by studying quasi-one-dimensional systems of ultracold atoms consisting of a single impurity interacting with an increasing number of identical fermions, and the deterministic preparation of antiferromagnetic Heisenberg spin chains consisting of up to four fermionic atoms in a one-dimensional trap. These achievements make the ultracold atoms an ideal platform to study many-body physics in a bottom-up approach, i.e., one starts from the fundamental building block of the system and observes the emergence of many-body effects by adding atoms one by one into the system. Corresponding theoretical models have been developed to explain the experimental data, to tackle the crossover boundary between few and many particles, and even explore the solvability and integrability of the models, especially the energy spectrum of interacting few atoms such as two atoms in a harmonic trap, two heteronuclear atoms of unequal mass in a ring trap, and two atoms in a $\delta$-barrier split double well potential. After a brief review of Bethe-Ansatz method, a theory for the tunneling of one atom out of a trap containing two interacting cold atoms is developed based on the calculation of the quasiparticle wave function, and the tunneling dynamics of two atoms starting from the NOON state is explored from the exactly solved model of $\delta$-barrier split double well based on a Bethe ansatz type hypothesis of the wave functions. It was shown that the spectroscopy and spin dynamics for strongly interacting few atoms of spin-1/2 and spin-1 can be described by effective spin chain Hamiltonians, which serves as a useful and efficient tool to study the quantum magnetism with clod atoms.