 Wave optics starts with Huygens (1629  1695) and Young (1773  1829); see the link
for some details. Wave optics proceeds in essentially two steps. First, the Huygens principle is applied, second
interference between the resulting waves is added. 
  Step 1: A wave hitting some edge or just about anything will produce a circular wave as shown below. The effect is easily visible when looking at water waves on some
relatively undisturbed water surface with obstacles. 
  (Plane) waves hitting an obstacle as shown below thus are
detectable even in "shaded" places. Note that the circular wave would go all the way around the edge; this
is not shown for simplicity. The consequences are clear: 
  Even sharp edges
appear always blurred if one looks closely enough. i.e. on a µm scale. This happens for any plane wave, be
it light, radio waves (allowing you to receive the rhythmic noise of your choice even in the "shade" of,
e.g., buildings), or electron waves (in electron microscopes). Just the lengths scales are different, going with the
basic wave lengths of the wave considered. 
 
 The blue lines show the maxima (or minima; whatever you prefer) of the amplitude of a traveling wave at
some point in time t_{0}. The distance between the lines is thus the wavelength l Somewhat later, the whole system of lines would have moved somewhat to the right (the
propagation direction here). The distance of one wavelength is covered after a time t_{l} = l/v; v is the propagation speed. The time
t_{l} is the inverse frequency n
Thus we have v = n · l = c for light. 

  The consequences are clear: The
resolution d_{min} of a lens with the numerical
aperture NA, i.e. its capability to image two points at a distance d_{min}
separately and not as some blur, is wavelength limited and given by 
 

  It's easy to see in a "handwaving" manner
why the numerical aperture comes in. Imagine some lens and reduce its numerical aperture by putting a real opaque aperture with a hole in front of it. The aperture edges will induce a blur that
get's worse the smaller the hole and therefore NA. Your resolution goes down with decreasing NA. 
  On the other hand, your lens aberrations become worse with increasing NA. The resulting conflict
for optimized optical apparatus is clear and encompasses a lot of intricate and very advanced topics in optics, e.g.
how to make structures on microelectronic chips with lateral dimensions around 30 nm <
l with optical lithography. 
 Now we consider step 2: two waves can
interfere with one another. The principle as shown below is clear. 
 

  For a phase
difference = 0 we have constructive interference; the
amplitudes are doubled. For a phase difference = 180^{0} = p destructive interference "cancels" the wave; the amplitude is
zero. 
  The apparent paradox of how you can get nothing from something (where are the two single waves and the energy
they carry now?) is not trivial to solve; for details see the link. 
 The paradigmatic experiment for showing interference
effects is, of course, the double slit experiment. If you consider that for
electron waves, and in particular just for one electron (or photon), you are smack
in the middle of quantum theory. 
  In the wave picture the two spherical (or here
cylindrical) waves emanating from the two slits interfere to give the pattern shown below. There is no problem at
all. 
  In the photon picture, a photon (or electron)
passing through the two slits interferes with itself. This boggles the mind quite a
bit but the result is the same: You get the interference pattern as shown, with pronounced minima and maxima of the
intensity, which now correspond to the probability of detecting the particle. 
 

 Whenever we look at nontrivial optics, we need to consider interference effects. In the real
world (as opposed to the ideal world shown in the pictures above), we need to consider the fact that our waves are
almost never monochromatic (all have the same wavelength) and coherent (all have the same phase) plane waves extending into infinity in every
direction. 
  The exception is, of course, the typical Laser beam, where we have an (almost) monochromatic and (almost)
coherent beam. However, a "Laser beam" is typically "thin" and doesn't extend in all directions.
So it is not a simple plane wave! 
 A first important conclusion can be arrived at. 
  If we look at an ensemble of waves with the same wavelength, or better: with the same wave vector k = 2p/l
since it contains in addition to the wavelength also the direction of
propagation (that's why it's a vector), we note that: 
 
An ensemble of sufficiently many waves with the same ±k and random phases interferes to exactly zero (plus some
noise) 

  "Random
phases" means that all phases are equally probable. The proof of the theorem is easy: If one of the
many waves has a phase a, there will be some other wave with the phase
–a  the two will cancel. A visual proof constructed in a different context
(that should be familiar) but fits just as well here is shown in the link. 
 From this you realize immediately that
inside some hollow tube of length L that reflects waves at either end, only waves with l = 2L/m; m = 1, 2, 3,... can exist. 
  Waves not meeting this criterion will, upon reflection at the end of the tube, produce a phaseshifted
wave with some new phase, twice this phase upon the second reflection and so on. Pretty soon you have waves with
random phases in there and—see above. 
  What that also means is: We
now have also all musical instruments covered, in fact everything where the term
"resonator" comes up. This includes also the free electron gas and to some extent electrons in a periodic potential as well as the Bragg condition for diffraction of waves at crystals. The list goes on. Quantum
theory deals with wave functions y after all, and the big difference to classical
physics comes from the simple fact that you let your wave functions interfere before you take the square, producing in
terms of probabilities the classical equivalent of a particle. 
  