This dissertation presents the results from five related studies of the nature of Lyman alpha emitting galaxies (LAEs) at high redshifts. These objects have long been predicted to be very young, and possibly the primordial first galaxies. Recent results have shown that these objects may in fact be more complicated. Three different studies at z ∼ 4.5 show that LAEs are mostly not primordial, as evidence was found for dust in most of our objects. These studies also show that not all LAEs are young, since two out of a sample of 15 LAEs appear to be much older (∼ 500 Myr). These older objects are detected with large Lyman alpha equivalent widths (EWs) at an older age due to a clumpy interstellar medium (ISM) geometry. This geometry results in a higher escape fraction for Lyman alpha photons than for continuum photons, thus enhancing the EW to arbitrarily high values. This effect may be responsible for the large number of high EW LAEs detected in recent studies. Extrapolating these stellar population results to the far-infrared (FIR), it appears that nearly half of our sample of LAEs could be detected in their rest-frame FIR with a future observatory such as the Atacama Large Millimeter Array. Detection of the dust emission from these galaxies could confirm the earlier results that many of these objects contain a significant amount of dust. The final study is concerned with LAEs at a much lower redshift of z ∼ 0.3. These LAEs are older, more massive, and more metal rich than their high-redshift counterparts, indicating evolution in this type of galaxy. Studying LAEs at low redshift is the highest priority for the future, as one can perform far more detailed analyses then those done at high redshift.