java.time

The java.util Date-Time API and their formatting API, SimpleDateFormat are outdated and error-prone. It is recommended to stop using them completely and switch to the modern Date-Time API^*.

Also, quoted below is a notice from the home page of Joda-Time:

Note that from Java SE 8 onwards, users are asked to migrate to java.time (JSR-310) - a core part of the JDK which replaces this project.

Solution using java.time, the modern Date-Time API:

import java.time.ZoneId;
import java.time.ZonedDateTime;

public class Main {
    public static void main(String[] args) {
        // ZonedDateTime.now() is same as ZonedDateTime.now(ZoneId.systemDefault()). In
        // order to specify a specific timezone, use ZoneId.of(...) e.g.
        // ZonedDateTime.now(ZoneId.of("Europe/London"));
        ZonedDateTime zdtDefaultTz = ZonedDateTime.now();
        System.out.println(zdtDefaultTz);

        // Convert zdtDefaultTz to a ZonedDateTime in another timezone e.g.
        // to ZoneId.of("America/New_York")
        ZonedDateTime zdtNewYork = zdtDefaultTz.withZoneSameInstant(ZoneId.of("America/New_York"));
        System.out.println(zdtNewYork);
    }
}

Output from a sample run:

2021-07-25T15:48:10.584414+01:00[Europe/London]
2021-07-25T10:48:10.584414-04:00[America/New_York]

ONLINE DEMO

Learn more about the modern Date-Time API from Trail: Date Time.

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^{* For any reason, if you have to stick to Java 6 or Java 7, you can use ThreeTen-Backport which backports most of the java.time functionality to Java 6 & 7. If you are working for an Android project and your Android API level is still not compliant with Java-8, check Java 8+ APIs available through desugaring and How to use ThreeTenABP in Android Project.}

Check out DateTimeZone & Interval:

DateTime dt = new DateTime();
    // translate to London local time
    DateTime dtLondon = dt.withZone(DateTimeZone.forID("Europe/London"));

Interval:

Interval interval = new Interval(start, end); //start and end are two DateTimes

I want to convert the current time to the time in a specific timezone with Joda time.

It's not really clear whether you've already got the current time or not. If you've already got it, you can use withZone:

DateTime zoned = original.withZone(zone);

If you're just fetching the current time, use the appropriate constructor:

DateTime zoned = new DateTime(zone);

or use DateTime.now:

DateTime zoned = DateTime.now(zone);