The relationships between four basic circuit variables - voltage (v), current (i), charge (q), and magnetic flux (ϕ) - have defined three fundamental circuit elements: resistor, capacitor, and inductor. From a symmetry view, there is a fourth fundamental circuit element defined from the relationship between charge and magnetic flux. Historically, a device called memristor was considered to be the fourth element, but it has caused intense controversy because the memristor is conceived based on a nonlinear i-v relationship rather than a direct q-ϕ relationship. Alternatively, a direct correlation between trapped charge (q) and magnetic flux (ϕ) can be built up by employing the magnetoelectric (ME) effects, i.e., magnetic field control of electric polarization and electric field control of magnetization. In this review, we summarize recent progress on the principle and applications of the fourth circuit element based on the ME effects. Both the fourth linear element and nonlinear memelement, termed transtor and memtranstor, respectively, have been proposed and experimentally demonstrated. A complete relational diagram of fundamental circuit elements has been constructed. The transtor with a linear ME effect can be used in a variety of applications such as the energy harvester, tunable inductor, magnetic sensor, gyrator, and transformer etc. The memtranstor showing a pinched hysteresis loop has a great potential in developing low-power nonvolatile electronic devices. The principle is to utilize the states of the ME coefficient αE=dE/dH, instead of resistance, magnetization or electric polarization to store information. Both nonvolatile memories and logic functions can be implemented using the memtranstors, which provides a candidate route toward the logic-in-memory computing system. In addition, artificial synaptic devices that are able to mimic synaptic behaviors have also been realized using the memtranstor. The fourth circuit element and memelement based on the ME effects provide extra degrees of freedom to broaden circuit functionalities and develop advanced electronic devices.