The magnetoelectric (ME) effects go back to the conjecture of Pierre Curie in 1894 that materials might exist that can be polarized electrically by means of a magnetic field and magnetized by means of an electric field. The different kinds of such ME effect known today can be classified on a thermodynamic and symmetry basis. This paper surveys the thermodynamically reversible Induced effects, i.e. the linear and bilinear ME effects, the piezomagnetoelectric and thermodynamically irreversible Spontaneous effects, i.e. electric field, magnetic field and stress induced switching or reorientation of domains with spontaneous polarization, spontaneous magnetization, spontaneous toroidal moment, spontaneous deformation. Examples of insulating materials displaying the different kinds of ME effect are discussed. Useful applications of ME effects are so far restricted to research: determination of magnetic symmetry, a powerful complementary tool to neuron diffraction, study of magnetic phase transitions, magnetic phase diagrams, toroidal moments, poling of antiferromagnetic domains, ME spin chirality control, linear magneto-optic effect, ME spectroscopy at optical frequencies.