Modular Towers are a special case of sequence of Hurwitz spaces (I usually take inner reduced, though other equivalences apply as well) defined by taking any collection of pairs \{(G_k,\bfC_k\}_{k=0}^\infty where the groups come with a collection of maps \psi_{k+1,k}: G_{k+1}\to G_k, \bfC_k is a collection of r_k conjugacy classes in G_k where the classes are counted with multiplicity, and  \psi_{k+1,k} maps \bfC_{k+1} to \bfC_k surjectively (taking account of multiplicity). 

The Modular Tower case for a prime p is the case with these attributes. 

1. G_0 is divisible by a prime p but has no \bZ/p quotient (is p-perfect).

2. Each G_k is the universal exponent p Frattini cover of G_0=G.  

3. \bfC_0 are a collection of p' classes of G_0. 

4. Each \bfC_k are the unique lifts to G_k of the classes of \bfC_0 (so r_k is constant in k, we just call it r). 

The corresponding Hurwitz spaces are often denoted simply \sH(G_k,\bfC)^* where * indicates the equivalence relation (absolute, inner, reduced).  

Why such restrictive conditions, instead of arbitrary generality? Because, in taking advantage of the Fried-Voeklein characterization of the R(egular)I(nverse)G(alois)P(roblem) in terms of rational points on Hurwitz spaces, this definition puts the RIGP in the position to generalize a famous result on modular curves, the Mazur-Merel result. The generalizing conjecture is called the Main Conjecture on MTs, and it just says in various forms we expect no K points at high tower levels no matter what number field is K. 

We know there can be more than one component of \sH(G_k,\bfC)^*, and it can be very difficult detecting components, especially at high levels. For example, Conway-Parker-Fried-Voelklein says something solid about what happens if you stick to one group, say G_0=G, fix a starting set of r^* distinct conjugacy classes \bfC^*, but consider all conjugacy classes \bfC with support in \bfC^*. (To simplify assume \bfC contains each class of \bfC^* at least once). Then, you can indicate the set \sC_{\bfC^*} of these by (\bN^+)^{r^*}. Then, Conway-Parker-Fried-Voelklein says there is a (n_1',\dots,n_{r^*}') so that if \bn\in \sC_{\bfC^*} then you can tell exactly what is the number of components of the Hurwitz space \sH_{\bn} and what are their definition fields.    

This result, however, tells nothing about the nature of the projective sequence of components on a Modular Tower, even if at level 0, you know there is just one component. The Main Conjecture is a triviality unless (on a given MT) there is a projective sequence of components defined over a fixed number field K. Progress in the use of Modular Towers is dependent on some other conjectures and results. Foremost among these. 

1. A classification of cusp types phrased using the Hurwitz monodromy group. 

2. Three "Frattini Principles" that translate the use of the Universal p-Frattini cover into properties of cusps. 

3. A cusp type (Harbater-Mumford) that has allowed many proving properties of Hurwitz spaces when they are present. Example: We know many MTs do have projective sequences of components over \bQ (say) by applying an easily testable criterion to certain H-M cusps.   

4. The g-p' conjecture that there is a projective sequence of K components for some K on a Modular Tower if and only if the cusp type called g-p' appears at a component at level 0. We know this is sufficient, and there is much evidence it is necessary.