Q:

How could one trade volatility skew if you think it's too flat or steep?

We all know that you can trade on a forecast of volatility by dynamically hedging, but I'm wondering if there's a similar technique where in you can trade the skew specifically? 
Let's say you travel back to 1985 before the market has started to price in skew, how could you take advantage of this? 

A:

Trading the skew is a common practice for traders specializing in options.  Let's say you have a 3M skew curve like the blue one below (where I have highlighted a few key strikes) but you think the correct skew curve is more like the red one

Let's further assume you unwilling to make a bet on the overall height (level) of the curve.  You just want to bet on the shape.
There is a monotonic relationship between option price and volatility, so to make your bet you can buy the cheap option on the left, and short the less cheap option on the right all while staying neutral to overall volatility.
Option traders track sensitivity of an option to volatility itself, using the word "Vega" to denote it.  In addition, they track sensitivity to underlying price, called "Delta".  You can make yourself a table of the three interesting options on the original blue curve
Side  Strike K Price  ImpVol   Delta  Vega
 Put    125       2     38      -15    20
Call    140       5     30       50    100
Call    155       1     32       10    12

and then find a combination of options that is "vega neutral".  In this case it is easy to see you could sell 500 of the K=155 calls and buy 300 of the K=125 puts (for a total cashflow of -100) and have the vega neutral portfolio you desire.  (To be delta neutral, you also have to sell 5 of the underlying).
Note, however, that this bet sells the K=155 calls you find to be underpriced.  Not a good idea.  So you would actually prefer to buy 1000 of the K=155 calls and also buy 600 of the K=125 puts.  At the same time, you sell 120 of the K=140 calls, and this time you are vega neutral with all bets on the correct side.
Once the crash happens, and everybody realizes the true skew in red, you will profit.

