Unfortunately chilli plants and Chillis can be susceptible to a few different bugs and diseases, there is a huge amount of information around the internet so i'll tell you a bit about the most common ones below

Slugs and Snails- Probably the most common and most irritating little buggers in your garden, they sneak up in the middle of the night and wipe out all of your seedlings, destroy your fruit and much on the leaves of your lovely healthy chilli plants.

There are a number of ways to prevent them, I like to use organic, veg friendly pellets in the greenhouse and around the garden, I also partake in regular hunts disposing of any i find hiding under pots and shelving in the greenhouse. Other methods of prevention are slug traps and copper tape.

Greenfly - You can almost guarantee that at some point you will be inundated with these. They cover your plants leaves and suck all of the sap and goodness from them leaving them distorted and wilted, eventually killing your plants.

I generally combat them using marigolds, these work as a sacrifcial plant, the flies are attracted to them more than your chilli plants. They are also cheap and chearful and add a bit of extra colour to your greenhouse. I have also been known to catch ladybirds and release them in the greenhouse to feast of the greenfly. Other people I know use organic spray, weak washing up liquid spray etc if they have a particularly bad infestation.

Mould and Mildew - There are a number of different types of mould and mildew these usually occur due to the plants being too damp, high humidity or their not being enough air circulating around the chilli plants. There are also a few bacterial based moulds and diseases, i've never experienced any of them so fingers crossed you won't.