Why is the enthalpy change not zero in an adiabatic process? Which of the following must be true for adiabatic processes?
  
  
  $C_V = C_p$
  $\Delta H = 0$
  $\Delta U = 0$
  $\Delta S = 0$ 
  $q = 0$
  


(Source: Chemistry GRE)

The answer is $q = 0$. From what I can find, an adiabatic process is when there is no transfer of heat, but then why is the enthalpy change not zero?
The Enthalpy $H$ is defined as $H=U+PV$. Therefore, 

$$\Delta H=\Delta U + P\Delta V +V\Delta P$$

For an adiabatic process, $q=0$. therefore from the first law of thermodynamics, 
$$\Delta U = q +w =q-P\Delta V$$
$$\Delta U=w=-P\Delta V$$
Substituting this in the first equation you get, 
$$\Delta H=V\Delta P$$
If $\Delta P$ is zero during the process (Isobaric) then the enthalpy change will indeed be zero, but it will be non-zero if the process is not isobaric.