Hybrid beamforming is an attractive solution to build cost-effective and energy-efficient transceivers for millimeter-wave and terahertz systems. However, conventional hybrid beamforming techniques rely on analog components that generate a frequency flat response such as phase-shifters and switches, which limits the flexibility of the achievable beam patterns. As a novel alternative, this paper proposes a new class of hybrid beamforming called Joint phase-time arrays (JPTA), that additionally use true-time delay elements in the analog beamforming to create frequency-dependent analog beams. Using as an example two important frequency-dependent beam behaviors, the numerous benefits of such flexibility are exemplified. Subsequently, the JPTA beamformer design problem to generate any desired beam behavior is formulated and near-optimal algorithms to the problem are proposed. Simulations show that the proposed algorithms can outperform heuristics solutions for JPTA beamformer update. Furthermore, it is shown that JPTA can achieve the two exemplified beam behaviors with one radio-frequency chain, while conventional hybrid beamforming requires the radio-frequency chains to scale with the number of antennas to achieve similar performance. Finally, a wide range of problems to further tap into the potential of JPTA are also listed as future directions.