There are lots of illnesses that hamsters can develop over their life, here are some of the most common: 

• Colds- they are much like what we experience. The fur around the nose may seem messy and there may be nasal discharge. Sniffing and sneezing will also be heard. 
• Wet tail- this is a bacterial infection and often proves fatal. Stress is a big reason to as why they can develop this as the first few days and weeks in their new home can be nerve racking. You'll notice an odour, diarrhoea and walking with a hunched back along with many other symptoms. 
• Diarrhoea- can be confused with Wet Tail as they are very similar. Commonly a result of over feeding vegetables and fruits. You will notice this around your hamsters anus and tail, or from the faeces on the floor. 
• Constipation- a wet tail can be an indication. It's related to the food and water they are eating and drinking. Make sure there is fresh water and provide careful amounts of fruit and veg. 
• Mange- if your hamster shakes, scratches its ears and looses their fur they may have mange. Results in grey, warty scabs on the nose, ears and genitals. Clean their cage thoroughly and take to the vet immediately. 
• Fleas and Lice- passed on from cats, dogs, rats and mice. Take your hamster to the vet before buying any off the shelf treatment. Clean out the cage thoroughly. Symptoms are very similar to mange. 
• Runny eyes- tears may form in the eyes indicating trouble in the cheek pouches. Seek a vet's attention. 
• Lumps- any lumps may be cancerous tumours, and should be checked immediately.