Movable antenna (MA) technology offers promising potential to enhance wireless communication by allowing flexible antenna movement. To maximize spatial degrees of freedom (DoFs), larger movable regions are required, which may render the conventional far-field assumption for channels between transceivers invalid. In light of it, we investigate in this paper MA-enabled near-field communications, where a base station (BS) with multiple movable subarrays serves multiple users, each equipped with a fixed-position antenna (FPA). First, we extend the field response channel model for MA systems to the near-field propagation scenario. Next, we examine MA-aided multiuser communication systems under both digital and analog beamforming architectures. For digital beamforming, spatial division multiple access (SDMA) is utilized, where an upper bound on the minimum signal-to-interference-plus-noise ratio (SINR) across users is derived in closed form. A low-complexity algorithm based on zero-forcing (ZF) is then proposed to jointly optimize the antenna position vector (APV) and digital beamforming matrix (DBFM) to approach this bound. For analog beamforming, orthogonal frequency division multiple access (OFDMA) is employed, and an upper bound on the minimum signal-to-noise ratio (SNR) among users is derived. An alternating optimization (AO) algorithm is proposed to iteratively optimize the APV, analog beamforming vector (ABFV), and power allocation until convergence. For both architectures, we further explore MA design strategies based on statistical channel state information (CSI), with the APV updated less frequently to reduce the antenna movement overhead. Simulation results demonstrate that our proposed algorithms achieve performance close to the derived bounds and also outperform the benchmark schemes using dense or sparse arrays with FPAs.