The answers provided so far have provided some insight into the challenge of the question as posed. But it's not a challenging question. It just has to be posed properly; that is to say, it should be stated not in terms of macroscopic charge and current, but in terms of electromagnetic force.
First, to quickly define the notion of the electric and magnetic fields. Early observations in electricity showed charged objects (e.g. the famous rubbing of amber, causing the amber to pick up dust and things) attracting and repelling one-another, without being in contact, thus the electric force was defined. Magnetism was of course observed as a force between something like lodestone and iron ore. Currents produce similar force. Thus we have magnetic force. It is useful to abstract the forces, for mathematical reasons, to vector fields. The electric field is the force per charge at a point in space, and is thus represented by a force vector. The magnetic field is dependent on current, the macroscopic flow of charge we now know, and in fact the force on an object is not absolute, but depends on the movement of the object. All force is restricted to one plane of force, and the magnetic field vector defines this plane by pointing normal to it with a magnitude indicating the magnitude of the force per charge per velocity. Thus the magnetic field vector is not a force vector, but an area vector, representing the plane of forces possible.
Okay, we now have those cleared up. From Maxwell's equations in differential (point) form in free space, that is, in the absence of charge or current densities, we arrive at the Helmholtz wave equation,
$$ \nabla^2 \mathbf{E} = \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} $$
Which, as has appeared at various points in mathematical physics, is a 3D wave equation. It holds any time $\rho = 0, \mathbf{J}=0$. That is there is no charge or current at a given point. An analogous equation holds for $\mathbf{H}$, the magnetic field.
That means every electromagnetic field obeys a 3D wave equation when in free space. This includes electrostatic solutions, electrodynamic solutions, everything in free space.
While understanding what the wave equation says is nice and enlightening to some extent, the point I'm hoping to make is that the idea of the electromagnetic field is almost like a pool of water. Every change causes a "wave" in that it is a solution to the wave equation. It essentially implies that changes in the electromagnetic field are not instantaneous, but propagate at speed $c$ (the speed of light). So what causes electromagnetic waves? Anything that results in electromagnetic fields in space. That is, any current or charge distribution (macroscopically speaking).
By making simplifying assumptions, most often by assuming that at any given point the direction of wave propagation is well defined, and perhaps by imposing boundary conditions, we can make a very insightful study of electromagnetic wave propagation in terms of sinusoidal waves in space and time. Combining this with Fourier analysis gives a convenient formulation of electromagnetic waves in terms of frequency content, which can be helpful. We also discover the requirements for the propagation of different propagation "modes", where a propagation mode can be considered any arrangement of the fields which produces changes in time according to the real part of $e^{j\omega t}$ at each point and changes in space according to the real part of $e^{-j\beta z}$ for propagation in the $\hat{z}$ direction with phase/propagation constant $\beta$ ($\gamma$ is the general propagation constant, with real and imaginary/phase components). This means that not only does the field at any given point change magnitude and direction cyclically, but the same thing occurs as we move in space.
With such a simplification, the wave equation above gives a number of special cases, namely transverse electromagnetic (TEM) waves, with the electric and magnetic field vectors both transverse to propagation direction, as well as transverse electric (TE) and transverse magnetic (TM) waves, which must propagate in between particular boundaries to exist. We can also look at spherical waves, which are quite common, as you can imagine. Even among these special cases there are further special cases, which allow us to calculate exact field solutions analytically (without computer calculation, numerical solutions) for some geometries. This is really the domain of radio (RF), microwave, and optical engineers and scientists.
However, it is the antenna engineers and scientists who study the radiation of electromagnetic waves. I do a lot of microwave frequency work, and I thought I had it bad with vector calculus, but the classical treatment of antennas is downright tedious.
The topic of radiation, then, focuses on what happens at an interface between e.g. a conductor and free space. Moreover, what exactly do the distant fields look like, and how efficient is the transfer of power/energy? That's a harder question. It's best tackled using the scalar and vector potentials and a lot of vector calculus. That's how it was done by Maxwell, and thats how it's taught today, there's not much getting around it. It's a difficult topic.
All this comes about in terms of classical electrodynamics, which was developed mostly in the 19th century by mathematical physicists and mathematicians including Laplace, Poisson, Green, Gauss, Hamilton, d'Alembert, and Maxwell. Experimentally, Hertz was the first to demonstrate electromagnetic waves. It's worth noting, however, that even Hertz, performing the first experiments in EM wave behavior, noticed the inability of classical electrodynamics to explain some phenomena (in this case, the photoelectric effect). So in essence, when you look closer, it's worth modifying classical electrodynamics into QED and relativistic classical electrodynamics. This is where the idea of charge falls apart, as mentioned by previous answers. Anyway, I hope this is helpful. Just remember that the true insight is in the mathematics, not the stories we tell about the mathematics.
