How do I convert a dictionary to a JSON String in C#?

Json.NET probably serializes C# dictionaries adequately now, but when the OP originally posted this question, many MVC developers may have been using the JavaScriptSerializer class because that was the default option out of the box.

If you're working on a legacy project (MVC 1 or MVC 2), and you can't use Json.NET, I recommend that you use a List<KeyValuePair<K,V>> instead of a Dictionary<K,V>>. The legacy JavaScriptSerializer class will serialize this type just fine, but it will have problems with a dictionary.

Documentation: Serializing Collections with Json.NET

This answer mentions Json.NET but stops short of telling you how you can use Json.NET to serialize a dictionary:

return JsonConvert.SerializeObject( myDictionary );

As opposed to JavaScriptSerializer, myDictionary does not have to be a dictionary of type <string, string> for JsonConvert to work.

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Runtime.Serialization.Json;
using System.IO;

namespace ConsoleApplication1
{
    class Program
    {
        static void Main(string[] args)
        {
            Dictionary<int, List<int>> foo = new Dictionary<int, List<int>>();

            foo.Add(1, new List<int>( new int[] { 1, 2, 3, 4 }));
            foo.Add(2, new List<int>(new int[] { 2, 3, 4, 1 }));
            foo.Add(3, new List<int>(new int[] { 3, 4, 1, 2 }));
            foo.Add(4, new List<int>(new int[] { 4, 1, 2, 3 }));

            DataContractJsonSerializer serializer = new DataContractJsonSerializer(typeof(Dictionary<int, List<int>>));

            using (MemoryStream ms = new MemoryStream())
            {
                serializer.WriteObject(ms, foo);
                Console.WriteLine(Encoding.Default.GetString(ms.ToArray()));
            }
        }
    }
}

This will write to the console:

[{\"Key\":1,\"Value\":[1,2,3,4]},{\"Key\":2,\"Value\":[2,3,4,1]},{\"Key\":3,\"Value\":[3,4,1,2]},{\"Key\":4,\"Value\":[4,1,2,3]}]

Serializing data structures containing only numeric or boolean values is fairly straightforward. If you don't have much to serialize, you can write a method for your specific type.

For a Dictionary<int, List<int>> as you have specified, you can use Linq:

string MyDictionaryToJson(Dictionary<int, List<int>> dict)
{
    var entries = dict.Select(d =>
        string.Format("\"{0}\": [{1}]", d.Key, string.Join(",", d.Value)));
    return "{" + string.Join(",", entries) + "}";
}

But, if you are serializing several different classes, or more complex data structures, or especially if your data contains string values, you would be better off using a reputable JSON library that already knows how to handle things like escape characters and line breaks. Json.NET is a popular option.