How to create an instance of anonymous class of abstract class in Kotlin?

From the official Kotlin language documentation:

window.addMouseListener(object : MouseAdapter() { 
    override fun mouseClicked(e : MouseEvent) { 
    // ... 
}

Applied to your problem at hand:

val keyListener = object : KeyAdapter() { 
    override fun keyPressed(keyEvent : KeyEvent) { 
    // ... 
} 

As Peter Lamberg has pointed out - if the anonymous class is actually an implementation of a functional interface (i.e. not of an abstract class), SAM Conversions can be used to simplify this statement even further:

val keyListener = KeyAdapter { keyEvent ->
    // ...
}

Please also note this discussion about the different usage of interfaces defined in Java and Kotlin.

In Java

private Comparator<Integer> comparator = new Comparator<Integer>(){
    @Override
    public int compare(Integer o1, Integer o2) {
        return o1.compareTo(o2);
    }
};

in Kotlin

private val comparator = object: Comparator<Int> {
    override fun compare(o1: Int, o2: Int): Int {
        return o1.compareTo(o2)
    }
}

But because Comparator is a FunctionalInterface we can take advantage of SAM conversions and write

private val comparator = Comparator<Int> { o1, o2 -> o1.compareTo(o2) }

Please note that SAM feature only works for Functional interfaces defined in java, if you need to create object of a kotlin interface with single method then you will have to use object syntax.